GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
ANNA CADORET AND ARNO KRET
Abstract: A réécrire.
2010 Mathematics Subject Classification. Primary: 11G18, 14G35; Secondary: 20E42.
1. Introduction
Let Af denote the ring of finite adeles of Q. Let (G, X) be a Shimura datum and Sh(G, X) the associated
Shimura variety. For a neat compact open subgroup K ⊂ G(Af ) let ShK (G, X) := Sh(G, X)/K denote
the associated finite level of Sh(G, X). Fix s ∈ Sh(G, X). Let sK ∈ ShK (G, X) denote its image in
ShK (G, X) and S ⊂ ShK (G, X) the geometrically connected component (defined over a finite abelian
extension ES of the reflex field E of (G, X)) containing sK . To these data we associate an adelic
representation
ρs : π1 (S) → K ⊂ G(Af )
of the étale fundamental group1 of S. This representation - up to K-conjugacy - only depends on the
connected component of Sh(G, X)ES over S. So we will simply denote it by ρ : π1 (S) → K ⊂ G(Af )
in the following. For every prime `, let ρ` : π1 (S) → K ⊂ G(Af ) → G(Q` ) denote the corresponding `-adic component. Eventually, for every t ∈ S, regarded as a morphism t : spec(k(t)) → S, let
σt : Galk(t) → π1 (S) denote the morphism induced by functoriality of étale fundamental group (here,
k(t) denote the residue field at t). We say that t ∈ S is Galois-generic (resp. `-Galois-generic) (with respect to ρ) the image of ρ◦σt is open in the image of ρ (resp. the image of ρ` ◦σt is open in the image of ρ` ).
In this note, we discuss the existence and abundance of Galois-generic points in the above sense following
two independent approaches:
- The (`-GG ⇔ GG) problem;
- Generalized Hecke orbits.
Before reviewing our main results, let us mention that, in the context of Shimura varieties, the terminology ’Galois-generic’ was originally introduced by Pink in [P05]. The definition of Pink ( [P05, Def.
6.3]) does not resort to the formalism of étale fundamental groups and is seemingly stronger than ours.
Namely, if E ab denotes the maximal abelian extension of E, a point t ∈ S is Galois-generic in the sense
of Pink if and only if the induced point tE ab ∈ SE ab is Galois-generic in our sense. To show that the
two definitions are equivalent, we have to check that every open subgroup of the image of the geometric
étale fundamental group has finite abelianization. This is ensured by the fact that this geometric image
coincides with the adelic closure of K ∩ G(Q) in G(Af ) and a general group-theoretical result (Lemma
5.5): if G is a semisimple algebraic group over Q and Γ ⊂ G(Q) an arithmetic subgroup, then every open
subgroup of the adelic closure Γ− of Γ in G(Af ) has finite abelianization.
We hope our description of Galois-generic points on Shimura varieties in terms of representation of the
étale fundamental group will give new insights about their properties.
1As usual, we omit fiber functors in our notation for étale fundamental groups - see Remark 5.1
1
2
ANNA CADORET AND ARNO KRET
ρ
Let us also point out that the Galois representations ρ ◦ σt : Galk(t) → π1 (S) → K coincide - up to
K-conjugacy - with the ones introduced in [UY14].
1.1. The (`-GG ⇔ GG) problem. The first question which arises is whether there exists Galois-generic
points (other than the generic point) on S. Let S`gg , S gg denote the set of `-Galois-generic points and
gg
Galois-Generic points respectively. Write also S∞
:= ∩` S`gg and S gg∞ := ∪` S`gg . While `-adic specialization techniques (see Subsection 3.3.3) give rise to ’lots of’ `-Galois-generic points, they usually fail to
gg
ensure that S∞
(hence a fortiori S gg ) contains a point other than the generic point of S.
On the other hand, for adelic representations attached to motives, the `-adic Tate conjectures predict
gg
that the equality S∞
= S gg∞ should hold and modulo-` variants of the Tate conjectures predict that the
gg
gg
equality S = S∞ should hold as well. We refer to Subsection 3.3 for more details.
Using that partial forms of the `-adic and modulo-` Tate conjectures are available for Abelian varieties
( [FW84]), it can be shown that the equality S gg = S gg∞ holds for adelic representations attached to the
Tate module of Abelian schemes ( [C15]). Building on this result, it is not difficult to deduce by standard
arguments (moduli description of Siegel Shimura varieties and functoriality) that the equality S gg = S gg∞
holds for adelic representations attached to Shimura varieties of Hodge type. Using that our definition of
Galois-Generic points coincide with the one of Pink and a further group-theoretical ingredient (Lemma
6.3), we are able to go one step further and prove:
Theorem 1.1. Assume that (G, X) is a Shimura datum of abelian type. Then, with the above notation,
every `-Galois-generic points with respect to ρ : π1 (S) → K ⊂ G(Af ) is Galois-generic.
The adelic representations attached to Shimura varieties of Abelian type are known to be motivic so
Theorem 1.1 is in the realm of the modulo-` Tate conjectures.
This leaves the case of Shimura data (G, X) for which G has a factor of type E6 , E7 or mixte type D
aside. The adelic representations attached to these Shimura data are not known to be motivic and though
we suspect that Theorem 1.1 should extend to them, we are neither able to prove it nor even to show
that, in these cases, there exists Galois generic points other than the generic point on S.
1.2. Generalized Hecke orbits. A consequence of Theorem 1.1 is that, on Shimura varieties of Abelian
type, there exists ‘lots of’ Galois-generic points (in particular closed ones) ‘in an arithmetical sense’. More
precisely, as already mentioned, for `-Galois-generic points, we have at disposal abundance results of arithmetic nature . For instance, say that an irreducible curve C ,→ S is Galois-generic if its generic point
is. Then one can show that the set of irreducible Galois-generic curves defined over a number field is
Zariski-dense in S and that, for each such curve C ,→ S, defined over a finite extension say EC of ES
and integer d ≥ 1, all but finitely many closed points t ∈ C with [k(t) : EC ] ≤ d are Galois-generic.
For representations attached to connected Shimura varieties, the Zariski-density of closed Galois-generic
points is not surprising once the existence is proved: it follows from the facts that being Galois-generic is
preserved by Hecke operators and that Hecke orbits are Zariski dense. But the restrictions on the degree
are interesting. Indeed, it follows from the definition of Galois-generic points and the group theoretical
fact that there are only finitely many Hecke operators of bounded degree on a connected Shimura variety
(see Theorem 7.3) that for every Galois-generic point t ∈ S and integer d ≥ 1 there are only finitely
many t0 in the Hecke orbit of t with [k(t0 ) : E] ≤ d. Thus there are infinitely many Hecke orbits of
closed Galois-generic points on connected Shimura varieties of Abelian type and even, infinitely many
Hecke orbits of closed Galois-generic points intersecting a Galois-generic curve defined over a number field.
Using equidistribution techniques, we can strengthen these results as follows.
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
3
Let (G, X) be a Shimura datum (we no longer assume it is of Abelian type). Let K ⊂ G(Af ) be a neat
compact open subgroup. Write Γ := K ∩ G(Q)+ and let X + ⊂ X be a connected component defining a
connected component S ⊂ ShK (G, X). Write Aut(G, X + ) for the group of automorphisms of G defined
over Q and stabilizing X + . For every s ∈ S, write
[
b
Tφ (s),
Θ(s)
:=
φ∈Aut(G,X + )
for the (full) generalized Hecke orbit of s, where Tφ denotes the generalized Hecke operator induced by φ
on S (Subsection 7.1). Then, we prove the following case of the André-Pink Conjecture ( [A89, Ch. X,
Pb. 3], [P05, Conj. 1.6], [O13, Conj. 1.3]).
Theorem 1.2. (André-Pink Conjecture for Galois-generic points) Assume that G is almost Q-simple.
b
For every Galois-generic point s ∈ S, every infinite subset of Θ(s)
is Zariski-dense in S.
For instance, a consequence of Theorem 1.2 for Shimura varieties of Abelian type is that if C ,→ S is
an irreducible Galois-generic curve defined over a number field, then C is cut by infinitely many Hecke
orbits of closed Galois-generic points and each of these Hecke orbits cuts C in only finitely many points.
Theorem 1.2 extends a previous result of Pink ( [P05, Thm. 7.6]) for the Siegel Shimura varieties; our
proof follows the one of Pink but with some technical adjustments required to deal with non simply
connected groups G. More precisely, the main ingredient in Pink’s argument is an equidistribution result
of Clozel, Oh and Ullmo ( [ClOU01, Thm. 1.6]) which asserts that if G is connected, simply connected,
almost Q-simple with G(R) non-compact and if Γ ⊂ G(Q) is a congruence subgroup then for every sequence a = (an ) ∈ G(Q) such that lim |Γ\Γan Γ| = +∞ and y ∈ Γ\G(R), the Hecke orbits Tan (y), n ≥ 0
n→∞
are equidistributed on Γ \ G(R) (actually, Pink uses a variant of this result [ClOU01, Thm. 1.6, Rem (3)]
for GSp2g ). To deal with arbitrary Shimura varieties, one needs a generalization of this equidistribution
result for adjoint groups G and arithmetic (not only congruence) subgroups Γ ⊂ G(Q). Such a generalization was proved by Eskin-Oh ( [EO06, Thm 1.2]) following an idea of Burger-Sarnak ( [BuS91]) — see
Section 7.2. However, to apply Eskin and Oh’s result to our situation, we have to ensure that, for an
arithmetic subgroup Γ ⊂ G(Q), there are only finitely many Hecke operators Ta , a ∈ G(Q) with |Γ \ ΓaΓ|
bounded (Theorem 7.3). We provide a detailed proof of this result in the appendix. We proceed in
two steps. First, we prove the adelic variant of Theorem 7.3 (Proposition 8.5); here the ‘natural’ tool
are avatars of the Bruhat-Tits decomposition, which give explicit estimates for the local degrees (see
Subsection 8.2 for details). Then, we deduce Theorem 7.3 from this adelic variant by reducing to the
simply connected case, where we can apply strong approximation.
After a first version of this paper was released, we were informed by Hee Oh that Theorem 7.3 could also
be proved by equidistribution arguments as the ones used in [EO06] (see Remark 8.1 for details).
If we restrict to connected Shimura varieties S of Abelian type, Theorem 1.2 can easily be recovered from
Orr’s thesis ( [O13, Thm. 1.5 (ii)]) arguing as follows. Let s ∈ S be a Galois-generic point, let A be an
b
infinite subset of Θ(s),
and let Z be some irreducible component of the Zariski-closure of A containing
s. Since s is Galois-generic, it is Hodge-generic (Lemma 5.3). Hence the smallest special subvariety of
S containing s is equal to the smallest special subvariety of S containing Z (because in both cases, this
subvariety has to be S itself). By construction A ∩ Z is Zariski-dense in Z hence, from [O13, Thm. 1.5
(ii)], Z is weakly special. But as Z contains the Hodge-generic point s, this forces Z = S. Compared
with our result, [O13, Thm. 1.5 (ii)] relies on different techniques (isogeny bounds for Abelian varieties
and o-minimality) and is more general since it works for Hodge-generic points but, as stated, only apply
to Shimura varieties of Abelian type.
∗∗∗
The paper is organized as follows. In Section 3, we define Galois-generic points attached to adelic
representations of the étale fundamental group of a scheme and review their basic properties; we also
discuss there in more details the (`-GG ⇔ GG) problem. In Section 4, we recall definitions and facts
4
ANNA CADORET AND ARNO KRET
about Shimura varieties. In Section 5, we construct the adelic representations attached to connected
Shimura varieties, review some of their basic properties and discuss the notion of Galois-generic and
Hodge-generic points in this particular setting. Section 6 is devoted to the proof of Theorem 1.1 and
Section 7 to the proof of Theorem 1.2. Eventually, the Appendix Section 8 is devoted to the proof of
Theorem 7.3. It can be read independently of the rest of the paper.
Contents
1. Introduction
1
1.1. The (`-GG ⇔ GG) problem
2
1.2. Generalized Hecke orbits
2
2. Notation
5
3. Galois-generic points
6
3.1. Definition
6
3.2. Elementary properties of (`-)Galois-generic points
6
3.3. The `-GG ⇔ GG problem
7
3.3.1. `-Galois-generic points
7
3.3.2.
Motivic representations
8
3.3.3.
Conjecture 3.5 for adelic representations attached to Abelian schemes
9
3.3.4.
Conjecture 3.5 for adelic representations attached to Shimura varieties
9
4. Shimura varieties
9
4.1. Shimura varieties and connected Shimura varieties
9
4.2. Canonical models
11
5. Adelic representations attached to Shimura data
11
5.1. Construction
11
5.2. Elementary properties
12
5.2.1.
The adelic representation attached to Siegel moduli varieties
12
5.2.2.
Functoriality
14
5.3.
6.
Galois-generic and Hodge-generic points, comparison with Pink’s definition
14
Conjecture 3.5 for adelic representations attached to connected Shimura data of Abelian type 16
6.1. Shimura data of Hodge type
16
6.2. Transfer of Galois-genericity
17
6.3. Shimura data of Abelian type
18
7. Equidistribution
18
7.1. Generalized Hecke operators
18
7.2. Equidistribution
20
7.3. End of the proof
20
8. Appendix: Degree of Hecke operators
21
8.1. Formal lemmas
22
8.2. The local and Adelic cases
23
8.2.1.
24
Review of Bruhat-Tits theory
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
5
8.2.1.1. The vectorial part of the extended affine Weyl group W v
25
c
8.2.1.2. The extended affine Weyl group W
25
8.2.1.3. The Weyl group W and the standard appartment A(S)
26
8.2.1.4. Parahoric and parabolic subgroups
26
8.2.1.5. The building I a (G, Qp )
26
8.2.1.6. Non type-preserving automorphisms
26
8.2.2.
Proof of Lemma 8.6
27
8.2.3.
Proof of Lemma 8.8
28
8.3. Proof of Theorem 7.3
29
References
31
Acknowledgement: This work began while the authors stayed at the Institute for Advanced Study;
they are very grateful to the Institute for its hospitality. The authors also thank Ziyang Gao for pointing out that Theorem 1.2 follows from Orr’s result in case of Shimura data of Abelian type and Akio
Tamagawa, for pointing out a gap in the proof of Theorem ??. The first author was partly supported
by the project ANR-10-JCJC 0107 from the ANR, the NSF grant DMS-1155. She thanks the Research
Institute for Mathematical Sciences (Kyoto). The second author thanks the Institute for Advanced Study
(Princeton), the Mathematical Sciences Research Institute (Berkeley), and the Max Planck Institute for
Mathematics (Bonn), where parts of this work were carried out. During his time at the IAS (resp. MSRI)
he was supported by the NSF under Grant No. DMS 1128155 (resp. No. 0932078000). Finally he thanks
CMLS at the École Polytechnique for a pleasant stay.
2. Notation
• The fields in this paper, when of characteristic 0, will always be assumed to be embedded into the field
C of complex numbers. For such fields, compositum, Galois, abelian, algebraic closures etc. will always
mean with respect to the given embedding into C.
• Given a field k, we will k for its separable closure and Galk := Gal(k|k) for its absolute Galois group.
• We use a superscript (−)◦ for connected components in the Zariski topology and (−)+ for connected
zar
components in other topologies. We use the notation (−)
for the Zariski-closure and the superscript
−
(−) for the closure in other topologies.
• Given an algebraic group G, we let Gder ⊂ G denote its derived subgroup, Z(G) ⊂ G its center,
pab : G → Gab := G/Gder its Abelianization and pad : G → Gad := G/Z(G) its adjoint quotient.
• Given schemes S and T over a field k, unless there is a risk of confusion, we write ST := S ×k T (that
is we omit the notation for the base field k). When T = spec(F ) for a field extension k ⊂ F , we write
SF := Sspec(F ) and we write S := Sk . However, when S =: s = spec(E) for a field extension k ⊂ E
of subfields and F is also embedded into C, we write sF := spec(EF ) that is, we implicitly pick the
connected component of s ×spec(k) spec(F ) corresponding to the embedding of E, F in C. Similarly,
we write s := spec(E) where, as before, E means the algebraic closure of E into C.
• Given a scheme S of finite type over a field k and a point s ∈ S, we write k(s) for the residue field (a
finitely generated extension of k) of S at s. Unless otherwise mention, we identify points s ∈ S and the
corresponding morphism of k-schemes s : spec(k(s)) → S. For every s ∈ S, let Fs denote the associated
fiber functor from étale covers of S to sets. Recall that, by definition, the étale fundamental group of S
based at s is the automorphism group π1 (S; s) of Fs and that, if S is connected, for every s, s0 ∈ S there
always exists isomorphisms of fiber functors α : Fs →F
˜ s0 and the set of functor isomorphisms Fs →F
˜ s0
6
ANNA CADORET AND ARNO KRET
˜ s0 (X)
is a π1 (S, s)-torsor. In particular, for every étale cover X → S, α yields a bijection α : Fs (X)→F
which is equivariant with respect to the isomorphism of étale fundamental groups
˜ 1 (S, s0 ),
π1 (S, s)→π
σ 7→ ασα−1 .
Thus, unless it helps understand the situation, we will omit the base-point s in our notation for étale
fundamental groups.
3. Galois-generic points
3.1. Definition. Let k be a field of characteristic 0 and let S be a smooth, separated and geometrically
connected scheme over k with generic point η. Let G be an algebraic group over Q and let
ρ : π1 (S) → G(Af )
be a continuous representation. For every prime `, write
Y
ρ
G(Q` ) → G(Q` )
ρ` : π1 (S) → G(Af ) ⊂
`
for its `-th component.
Every point s ∈ S, regarded as a morphism s : spec(k(s)) → S, induces by functoriality of étale fundamental group a morphism of profinite groups σs : Galk(s) → π1 (S) (which is a section of the canonical
projection π1 (Sk(s) ) Galk(s) ). Write
Πs := im(ρ ◦ σs ) 
Π`,s := im(ρ` ◦ σs )
/ Π := im(ρ) 


/ Π` := im(ρ` ) / G(Af )
/ G(Q` )
We say that s ∈ S is Galois-generic (resp. strictly Galois-generic) with respect to ρ if Πs ⊂ Π is open
(resp. if Πs = Π). Similarly, given a prime `, we say that s ∈ S is `-Galois-generic (resp. strictly
`-Galois-generic) with respect to ρ` if Π`,s ⊂ Π` is open (resp. if Π`,s = Π` ).
3.2. Elementary properties of (`-)Galois-generic points.
(1) As S is normal, η ∈ S is strictly Galois-generic.
(2) Let k ⊂ k̃ be a finitely generated field extension Then s ∈ S is Galois-generic with respect to ρ if and
only if sk̃ ∈ Sk̃ is Galois-generic with respect to ρ|π1 (Sk̃ ) . This follows from the commutativity of the
following diagram
Galk(s) o
Galk̃k(s)
sk̃
1
/ π1 (S )
s
k̃
/ π1 (S )
k̃
/ Gal
k̃
/1
1
/ π1 (S)
/ π1 (S)
/ Galk
/1
and the fact that the images of Galk̃k(s) → Galk(s) and π1 (Sk̃ ) → π1 (S) are open.
(3) As S is geometrically connected over k, we have a short exact sequence
1 → π1 (S) → π1 (S) → Galk → 1
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
7
σ
and as the image of Galk(s) →s π1 (S) → Galk is open in Γk , we see that s ∈ S is Galois-generic with
respect to ρ if and only if
Πs := Πs ∩ Π ⊂ Π := ρ(π1 (S))
is open. Another way to formulate this observation is the following. Let k ⊂ k̃ ⊂ k denote the (in
general infinite) Galois subextension corresponding to the image of ρ−1 (Π) ⊂ π1 (S) in Galk . Then
s ∈ S is Galois-generic with respect to ρ if and only if sk̃ ∈ Sk̃ is Galois-generic with respect to
ρ|π1 (Sk̃ ) .
Under additional assumptions, one can enlarge k̃. For instance, we have
Lemma 3.1. Assume that every open subgroup U ⊂ Π has finite abelianization. With the above notation, let k̃ ⊂ k̃ ab ⊂ k denote the maximal abelian extension of k̃ in k. Then s ∈ S is Galois-generic
with respect to ρ if and only if sk̃ab ∈ Sk̃ab is Galois-generic with respect to ρ|π1 (Sk̃ab ) .
Proof. The non trivial implication is the ’only if’ one. So assume that s ∈ S is Galois-generic with
respect to ρ. Then Πsk̃ ⊂ Π is open. In particular, it has finite abelianization. Thus its quotient
Πsk̃ Πsk̃ /Πsk̃ab , which is abelian being a quotient of Gal(k̃ ab |k̃), is finite. (4) If S 0 → S is a dominant morphism of finite type with S 0 connected (for instance a connected étale
cover)2 and s0 ∈ S 0 a point above s ∈ S then s ∈ S is Galois-generic with respect to ρ if and only if
s0 ∈ S 0 is Galois-generic with respect to ρ|π1 (S 0 ) .
(5) Let s ∈ S be a Galois-generic point and let U ⊂ Π be any open subgroup contained in Πs . Let SU → S
denote the connected étale cover corresponding to the open subgroup ρ−1 (U ) ⊂ π1 (S) and let k(s) ,→
k(s)U denote the finite field extension corresponding to the open subgroup (ρ ◦ σs )−1 (U ) ⊂ Galk(s) .
Then s ∈ S lifts to a k(s)U -rational point sU ∈ SU which is strictly Galois-generic with respect to
ρ|π1 (SU ) .
We also have also the following ’geometric’ variant of the above observation. Let U ⊂ Π be any open
subgroup contained in Πs . Let SU → S denote the connected étale cover corresponding to the open
subgroup ρ−1 (U ) ⊂ π1 (S). The étale cover SU → S extends to a connected étale cover SU → S ×k kU
defined over a finite extension kU of k. Up to replacing kU with kU k(s) and s with the kU k(s)-point
induced by s on SU ×kU kU k(s), we may assume that k(s) = kU and U := ρ(π1 (SU )) = Πs . Then,
as above, s ∈ S lifts to a kU -rational point on SU which is strictly Galois-generic with respect to
ρ|π1 (SU ) .
3.3. The `-GG ⇔ GG problem. We review here the main results and conjectures concerning the
existence of (`-)Galois-generic points when k is finitely generated of characteristic 0. Let S sgg ⊂ S gg ⊂ S
(resp. S`sgg ⊂ S`gg ⊂ S) denote the sets of (resp. `-)Galois-generic points with respect to ρ (resp. to ρ` ).
Clearly,
\ gg
gg
S gg ⊂ S∞
:=
S` ⊂ S gg∞ := ∪` S`gg .
`
3.3.1. `-Galois-generic points. For every prime `, we know that the set S`gg is non-empty and even ‘large
in an arithmetical sense’. More precisely, we have the following results.
(1) (Serre’s argument - See [S89, §10.6]) (Actually, here, we only need the assumption that k be Hilbertian).
Fact 3.2. There exists an integer d ≥ 1 and infinitely many closed strictly `-Galois-generic points
s ∈ S with [k(s) : k] ≤ d.
Proof. We recall the argument. For every prime `, as Π` is a compact `-adic Lie group, its Frattini
subgroup Φ(Π` ) ⊂ Π` is open. Let X` → S denote the connected étale cover corresponding to the
open subgroup ρ−1
` (Φ(Π` )) ⊂ π1 (S) and fix a finite étale cover π : S → U of degree - say d ≥ 1, where
U is a non-empty open subset of Pnk . Let X̂` → U denote the Galois closure of X` → U . Let M`
2S 0 is not necessarily geometrically connected over k but if k ⊂ k 0 denotes the finite field extension corresponding to the
image of π1 (S 0 ) by π1 (S) Galk then S 0 is geometrically connected over k0 .
8
ANNA CADORET AND ARNO KRET
denote the set of all subgroups of Gal(X̂` |S) which are inverse images of proper maximal subgroups
of Π` /Φ(Π` ) and for every M ∈ M` , let πM : SM → S denote the corresponding quotient. Then, by
Hilbert’s irreducibility theorem the set
[
Ũ := U (k) \
π ◦ πM (SM (k))
M ∈M`
is infinite. For every u ∈ Ũ and s ∈ π −1 (u), the morphism
ρ
σ
Γk (s) →s π1 (S) →` Π` → Π` /Φ(Π` )
σ
ρ
is surjective. Thus, by definition of the Frattini subgroup, the representation Γk (s) →s π1 (S) →` Π` is
surjective as well, that is s is strictly `-Galois-generic with respect to ρ` . (2) When k is finitely generated, S is a curve and ρ` (π1 (S)) has perfect Lie-algebra, we can improve Fact
3.2 as follows
Fact 3.3. ( [CT12], [CT13]) For every integer d ≥ 1 all but finitely many s ∈ S with [k(s) : k] ≤ d
are `-Galois-generic.
When S has higher dimension N , up to replacing S with a non-empty open subscheme, we can always
find a non-empty open subset S1 of PkN −1 with generic point η1 and a smooth, separated morphism
S → S1 with geometrically connected fibers. Applying Fact 3.3 (or Fact 3.2) to the base change Sη1
of S over k(η1 ), we get a Zariski-dense set of closed `-Galois-generic points on Sη1 hence a Zariskidense set of codimension 1 closed subschemes of S, smooth, separated, geometrically connectect over
a finite extension of k and whose generic point is Galois-generic with respect to ρ` . By induction
on codimension, we construct a Zariski-dense set of curves C ,→ S smooth, separated, geometrically
connected over a finite extension of k and whose generic point is Galois-generic with respect to ρ` .
Applying one more time Fact 3.3 to each of these curves we get the assertion of the introduction,
namely that
Corollary 3.4. The set of irreducible `-Galois-generic curves defined over a finite extension of k is
Zariski-dense in S and for each such curve C ,→ S, defined over a finite extension say kC of k and
integer d ≥ 1 all but finitely many closed points s ∈ C with [k(s) : kC ] ≤ d are `-Galois-generic with
respect to ρ` .
gg
3.3.2. Motivic representations. The above results are `-adic in nature and fail to ensure that S∞
contains
points other than the generic points. However, for adelic motivic representations, for instance, those of
the form
Y
ρ : π1 (S) →
GL(H∗ (Xη , Q` )),
`
where X → S is a smooth projective scheme, we expect that the following conjecture holds.
Conjecture 3.5. (`-GG ⇔ GG)
gg
S gg = S∞
= S gg∞ .
gg
More precisely, the equality S∞
= S gg∞ would follow from the Tate semisimplicity and fullness conjecgg
tures ( [S94], [A04, §7.3]) while the equality S gg = S∞
would follow from the modulo-` variant of the
Tate fullness conjecture formulated by Serre ( [S94, Conj. 10.2]) and the arguments of [C15].
For the Tate module of Abelian varieties, partial forms of the `-adic and modulo-` variants of the Tate
conjectures were proved by Faltings ( [FW84]). In the following, we will refer to these results as the
‘partial Tate conjectures for Abelian varieties’. These partial results are enough to prove Conjecture 3.5
for i = 1 and X → S an Abelian scheme as we recall now.
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
9
3.3.3. Conjecture 3.5 for adelic representations attached to Abelian schemes. Let A → S be an Abelian
scheme. As k has characteristic 0, for every integer N ≥ 1 the kernel A[N ] of multiplication-by-N on A
is an étale cover of S. Thus, for every s ∈ S, we get an adelic representation
Y
ρ : π1 (S; s) → GL(T (A)s ) '
GL(T` (A)s ),
`
n
where, T` (A) := lim
A[` ], T (A) :=
←−
Q
n
` T` (A)(=
lim
A[N ]) as étale sheaves on S.
←−
N
The following is a consequence of the partial `-adic Tate conjectures for Abelian varieties and the Borel-de
Siebenthal Theorem.
gg
Lemma 3.6. ( [H12]) S`gg = S∞
for every prime `, that is, if s ∈ S is `-Galois-generic for one prime `
then s ∈ S is `-Galois-generic for every prime `.
Using `-independance results for motivic representations (involving the Weil conjectures - [D74]), grouptheoretical lifting arguments and the modulo-` variant of the partial Tate conjectures for Abelian varieties,
we can improve Lemma 3.6 as follows.
gg
Theorem 3.7. ( [C15, Thm. 1.2]) S gg = S∞
that is Conjecture 3.5 holds for ρ : π1 (S; s) → GL(T (A)s ).
In particular, the abundance results for closed `-Galois-generic points in Subsection 3.3.1 (Fact 3.2, Fact
3.3, Corollary 3.4) also apply to closed Galois-generic points with respect to ρ : π1 (S) → GL(T (A)).
3.3.4. Conjecture 3.5 for adelic representations attached to Shimura varieties. In Section 6 we prove
(Theorem 1.1) Conjecture 3.5 for adelic representations attached to Shimura data (G, X) of Abelian type.
Shimura varieties of Abelian type are known to be moduli spaces for Abelian motives, at least under some
rationality conditions on the weight and center of the reductive group G (see for instance [Mi94, §3]). This
implies that the adelic representations attached to them are (closed to be) motivic (see the argument
for the Siegel Shimura varieties in Subsection 5.2.1). Our proof only uses the moduli description for
Siegel Shimura varieties, which is combined with the machinery of Shimura varieties, Galois categories
and group-theoretical Lemmas (see Section 6 for details). In particular, we do not need to impose further
rationality restrictions on the Shimura data of Abelian type (G, X). Still, we cannot handle the case of
Shimura data (G, X) which are not of abelian type - for instance those where G has a factor of type E6 ,
E7 or of mixte type D (see Subsection 6.3). Note that these representations are not known to be motivic.
However, in view of Theorem 1.1 it seems reasonable to expect that they satisfy Conjecture 3.5. On the
other hand, the moduli description of Shimura varieties of Abelian type suggests the search for a direct
proof of Conjecture 3.5 for abelian motives.
4. Shimura varieties
See for instance [D79], [Mo98, §1-2].
4.1. Shimura varieties and connected Shimura varieties. A Shimura datum is a pair (G, X) where
G is a connected reductive algebraic group over Q and X is the G(R)-conjugacy class of a morphism
x
of R-algebraic groups ResC|R Gm,C =: S →0 GR satisfying the following conditions (see [D79, (2.1.1.1),
(2.1.1.2), (2.1.1.3)]).
(D1) The space Lie(G)C viewed as a representation of C× via x0 and the adjoint action, only contains
the trivial character, the character z 7→ zz −1 and the character z 7→ zz −1 ;
(D2) Ad(x0 (i)) is a Cartan involution on Gad
R (where i ∈ C is a square root of −1);
ad
(D3) G has no Q-simple factor H on which the projection of x0 is trivial (equivalently, Gad is of
non-compact type that is, it has no Q-simple factor H with H(R) compact).
Under these assumptions, X carries a unique complex structure such that for every faithful finite dimensional Q-representation V of G the family of Hodge-structures defined by X is a variation of Hodge
structures and, for this complex structure, X is a symmetric hermitian domain ( [D79, Prop. 1.1.14, Cor.
1.1.16, 1.1.17]).
10
ANNA CADORET AND ARNO KRET
We also assume the following additional condition holds:
(4) G is the generic Mumford-Tate group of (G, X).
Recall that the generic Mumford-Tate group of (G, X) is the smallest algebraic subgroup M ⊂ G defined
over Q and such that every x : S → GR in X factors through MR . By construction M is normalized by
G(Q). As G(Q) is Zariski-dense in G ( [Bo76, Cor. 18.3]), M is normal in G. In particular, by Condition
(D3), M ad = Gad . This shows that X is again a M (R)-conjugacy-class and that (D2) and (D3), which
only involve Gad , hold. As Lie(M )C is a sub algebra of Lie(G)C , (D1) holds as well. So (M, X) is a
Shimura datum. Replacing (G, X) with (M, X) only affects the group of connected components of the
Shimura variety (more precisely, Sh(M, X) is a union of connected components of Sh(G, X) - see below).
In particular, it will not affect the representation of the étale fundamental group we will attach to it (see
Subsection 5). Hence, Condition (4) imposes no restriction on generality. From [UY14, Lemma 5.13],
Condition (4) implies
(4)’ Z(G)(Q) is discrete in Z(G)(Af ).
ad
ad
ad is
Let X ad denote the Gad (R)-conjugacy class of xad
0 := pR ◦ x0 : S → GR . The canonical map X → X
injective and identifies X with a union of connected components of X ad . Let X + denote the connected
component of x0 in X, which we identify with its image in X ad . The stabilizer of X + in Gad (R) is the
connected component Gad (R)+ of 1 and the stabilizer of X + in G(R) is the inverse image G(R)+ of
Gad (R)+ in G(R).
Given a subgroup Γ ⊂ G(Q), we write Γad ⊂ Gad (Q) for the image of Γ in Gad (Q). In the following, the
terminology ‘neat’ subgroup K ⊂ G(Af ) will refer to a subgroup K ⊂ G(Af ) which is neat in the sense
of [KY14, §4.1.4]. Every compact open subgroup of G(Af ) contains a neat compact open subgroup, every
G(Af )-conjugate and every open subgroup of a neat subgroup is again neat, if f : G2 → G1 is a morphism
of reductive groups over Q and K2 ⊂ G2 (Af ) is neat then f (K2 ) ⊂ G1 (Af ) is neat. If K ⊂ G(Af ) is
a neat subgroup Γ := K ∩ G(Q) ⊂ G(Q) is neat in the usual sense hence, in particular, torsion free.
Thus, by Condition (4)’, for every neat compact open subgroup K ⊂ G(Af ) we have K ∩ Z(G)(Q) = 1.
As a result, Γ := K ∩ G(Q)+ is isomorphic to Γad and we will often identify them in the following.
Furthermore, as (Γ ∩ Gder (Q))ad ⊂ Γad are both arithmetic subgroups of Gad (Q) ( [D79, Cor. 2.0.13]),
they are commensurable. This shows that Γ ∩ Gder (Q) is of finite index in Γ hence, up to replacing K
with a smaller open subgroup, we may and will assume that Γ ⊂ Gder (Q).
Let K ⊂ G(Af ) be a neat compact open subgroup and set
ShK (G, X)(C) := G(Q)\(X × (G(Af )/K))
'
G
Γg \X + ,
g∈G(Q)+ \G(Af )/K
where Γg := g −1 Kg ∩ G(Q)+ ⊂ G(Q)+ . The group Γg ⊂ G(Q)+ is a neat arithmetic subgroup. In
particular, Γg \X + is endowed with a canonical complex structure for which the quotient map X + →
Γg \X + is an analytic cover. By the Baily-Borel theorem ( [BBo66]), this complex structure is induced
by a canonical structure ShcΓg (G, X + )C of connected quasi-projective smooth complex algebraic variety
on Γg \X + = Γg \X + . The Shimura variety attached to (G, X, K) is
G
ShK (G, X)C :=
ShcΓg (G, X + )C .
g∈G(Q)+ \G(Af )/K
This discussion motivates the definition of connected Shimura varieties (See3 [D79, 2.1.8]): A connected
Shimura datum is a pair (G, X + ) where X + is the Gad (R)+ -conjugacy class of a morphism of R-algebraic
x
groups S →0 Gad
R satisfying Conditions (D1), (D2), (D3) above. For a neat compact open subgroup
K ⊂ G(Af ), the connected Shimura variety attached to (G, X + , Γ := K ∩ G(Q)+ ) is the quasi-projective
3Note that our definition differs slightly from Deligne’s definition in [D79, 2.1.8], where G is required to be semi-simple.
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
11
smooth complex algebraic variety ShcΓ (G, X + )C whose complex points are Γ\X + = Γad \X + .
For a neat compact open subgroup K and an open subgroup K 0 ⊂ K, the morphisms ShK 0 (G, X)C →
ShK (G, X)C are affine (even finite étale) hence the projective limit
Sh(G, X)C := lim
ShK (G, X)C
←−
K
exists as a C-scheme. Furthermore, for every g ∈ G(Af ), the morphism induced by right multiplication
·gK : ShK (G, X)C → Shg−1 Kg (G, X)C is an isomorphism of C-schemes. The collection of the ·gK when
K varies induces a right action of G(Af ) on Sh(G, X)C . The C-scheme Sh(G, X)C endowed with this
right action of G(Af ) is the complex Shimura variety attached to (G, X). Under Condition (4)’, we have
( [D79, Cor. 2.1.11]):
Sh(G, X)(C) = lim
ShK (G, X)(C) = G(Q)\(X × G(Af )).
←−
K
For a connected component
X+
⊂ X, the associated connected Shimura variety
Shc(G, X)C := lim
ShcΓ (G, X + )C
←−
Γ
identifies with the connected component
Sh(G, X)◦C
of Sh(G, X)C containing the image of X + × {1}.
The right action of G(Af ) induces a transitive action on the set of connected components π0 (Sh(G, X)C )
and makes it a torsor under the abelian quotient G(Af )/G(Q)−
+ ( [D79, Prop. 2.1.14]).
4.2. Canonical models. For a Shimura datum (G, X) let E(G, X) ⊂ C denote its reflex field ( [D79,
2.2.1]). Recall that E(G, X) is the field of definition of the G(C)-conjugacy class (independent of x0 )
of x0 ◦ µ : Gm,C → SC → GC , where µ : Gm,C → SC is defined on C-points by µ(z) = (z, 1). The field
E(G, X) is a number field (contained in the intersection of the splitting fields of G) and is functorial
in the pair (G, X): For every morphism of Shimura data (G2 , X2 ) → (G1 , X1 ) we have the inclusion
E(G1 , X1 ) ⊂ E(G2 , X2 ).
A canonical model of Sh(G, X)C is a scheme Sh(G, X) over the reflex field E := E(G, X) endowed with
a right action of G(Af ) and a G(Af )-equivariant isomorphism
Sh(G, X) ×E(G,X) C→Sh(G,
˜
X)C
such that the special points are algebraic and for a given type τ with corresponding dual field E ⊂ E(τ ),
the action of ΓE(τ ) ⊂ ΓE on the set of special points of type τ is the one given by the so-called reciprocity
map (see [D79, 2.2] for details).
By works of Deligne ( [D79]) and others (see for instance [Mo98, §2]), the Shimura variety Sh(G, X)C
has a (unique) canonical model Sh(G, X) over E(G, X) which is functorial in the pair (G, X): For every
morphism of Shimura data (G2 , X2 ) → (G1 , X1 ), the induced morphism of C-schemes Sh(G2 , X2 )C →
Sh(G1 , X1 )C descends uniquely to a morphism of E(G2 , X2 )-schemes Sh(G2 , X2 ) → Sh(G1 , X1 )×E(G1 ,X1 )
E(G2 , X2 ). For a connected Shimura variety ShcΓ (G, X + )C ⊂ ShK (G, X)C , let E(G, X) ⊂ EΓ (G, X + )
denote the finite field extension corresponding to the stabilizer of ShcΓ (G, X + )C under the action of
GalE(G,X) on the group of connected components π0 (ShK (G, X)C ). Note that E(G, X) ⊂ EΓ (G, X + ) is
an abelian extension. Then ShcΓ (G, X + )C descends to a geometrically connected component ShcΓ (G, X + )
of ShK (G, X) ×E(G,X) EΓ (G, X + ), which we call the canonical model of ShcΓ (G, X + )C over EΓ (G, X + ).
Again, these models are functorial in (G, X + , Γ) in the obvious sense.
5. Adelic representations attached to Shimura data
5.1. Construction. Let (G, X) be a Shimura datum, E := E(G, X) its reflex field. Fix s ∈ Sh(G, X)
and let X + ⊂ X denote the connected component such that the connected component of sC ∈ Sh(G, X)C
contains the image of X + ×{1}. Fix a neat compact open subgroup K ⊂ G(Af ). Write Γ := K ∩G(Q)+ ⊂
12
ANNA CADORET AND ARNO KRET
G(Q)+ for the corresponding arithmetic subgroup and EΓ := EΓ (G, X + ) for the field of definition of
SΓ := ShcΓ (G, X + ).
Let pK : Sh(G, X) → ShK (G, X) denote the canonical projections and write sK := pK (s). For every
open subgroup K 0 ⊂ K, write Γ0 := K 0 ∩ G(Q)+ and consider the (possibly non-connected) étale cover
SK 0 := ShK 0 (G, X) ×ShK (G,X) SΓ → SΓ .
Let S̃K 0 ⊂ SK 0 denote the connected component of sK 0 and let K̃ 0 ⊂ K denote the stabilizer of S̃K 0 in
K. Then define
ρs : π1 (SΓ ; sK ) → K ⊂ G(Af )
to be the representation attached to the projective system of Galois covers (S̃K 0 → SΓ )K 0 pointed by
s = (sK 0 )K 0 . In particular, its image is
\
Π := im(ρs ) =
K̃ 0 .
K0
The scheme S̃K 0 is connected but not geometrically connected over EΓ and SK 0 ×EΓ E splits into geometrically connected components isomorphic to SΓ0 ×EΓ0 E. In particular, the representation ρs,K 0 :
π1 (SΓ ; sK ) K̃ 0 /K 0 ⊂ K/K 0 attached to the Galois cover S̃K 0 → SΓ pointed by sK 0 restricts to a
surjective representation ρs,K 0 |π1 (S Γ ,sK ) : π1 (S Γ , sK ) Γ/Γ0 ⊂ K̃ 0 /K 0 . Passing to the limit, we get4
Π := ρs (π1 (S Γ ), sK ) = lim Γ/Γ0 = Γ− ⊂ K,
←−
0
where we identify limΓ/Γ with the adelic closure
←−
Γ−
of Γ in G(Af ).
Remark 5.1. If s0 ∈ Shc(G, X + ) is another point lying in the same connected component of Sh(G, X)EΓ
as s and with sK = s0K , then there exists k ∈ K such that s0 = sk. So we get ρs0 = k −1 ρs k. If we do not
assume that sK = s0K , it follows from the general formalism of Galois categories that there exists an (a non
unique) isomorphism of fibre functors α : FsK →F
˜ s0K such that α(SK 0 )(sK 0 ) = s0K 0 for all K 0 ⊂ K. Thus,
0
−1
if we let α̃ : π1 (S; sK )→π
˜ 1 (S; sK ), σ → ασα denote the induced isomorphism, tautologically, we have
ρs = ρs0 ◦ α̃. If we just pick an arbitrary isomorphism α : FsK →F
˜ s0K , then there exists kα ∈ K such that
0
−1
0
s = α(s)kα and we get ρs ◦ α̃ = k ρs k. As a result, we will often omit the subscript (−)s in our notation
for ρs : π1 (SΓ , sK ) → K ⊂ G(Af ) and refer to it as the adelic representation attached to (G, X + , K).
Also, for every open subgroup K 0 ⊂ K, as SΓ0 → SΓ is a connected étale cover, π1 (SΓ0 , sK 0 ) ⊂ π1 (SΓ , sK )
isan open subgroup and the representation attached to (G, X + , K 0 ) is just the restriction of the one
attached to (G, X + , K) to π1 (SΓ0 , sK 0 ) ⊂ π1 (SΓ , sK ). So changing the level is in general harmless. On
the other hand, note that, though G(Af ) acts transitively on π0 (Sh(G, X)C ), the dependency of ρs on
the geometrically connected component containing s is more subtile. We will not discuss it here since
our results are valid for an arbitrary choice of s ∈ Sh(G, X).
5.2. Elementary properties.
5.2.1. The adelic representation attached to Siegel moduli varieties. In this paragraph we explain that
the adelic representation attached to a Siegel Shimura variety coincides with the adelic representation
attached to the universal abelian scheme over it. For this, we review first the definition of Siegel Shimura
varieties.
Consider the Z-module Z2g equipped with the alternating pairing
g
g
X
X
hx, yistd =
xi yg+i −
xg+i yi .
i=1
i=1
Let GSp2n /Z be the algebraic group whose functor of points is given as follows. Let R be a ring, then
GSp2g (R) is the group of g ∈ GL2n (R) such that hg·, g·istd = ν(g)h·, ·istd for some ν(g) ∈ R× . Write
4Note in particular that the `-adic components ρ
s,` : π1 (SΓ , s) → G(Q` ) of ρs̃ : π1 (SΓ , s) → K ⊂ G(Af ) satisfy the
assumptions of Fact 3.3.
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
13
Ig ∈ GLg (Z) for the g × g-identity matrix, and Ag ∈ GL√
g (Z) for the g × g-matrix with zero’s everywhere,
except on the anti-diagonal, where we put 1’s. Choose −1 ∈ C and consider the morphism
√
aIg bAg
h(R) : S(R) −→ G(R), a + b −1 7→
,
−bAg aIg
for all R-algebras R. By definition X is the G(R)-conjugacy class of h. Then (G, X) is a Shimura datum
and it (resp. the associated Shimura variety) is called a Siegel Shimura datum (resp. a Siegel Shimura
variety). Since the group G is split, the reflex field of the datum (G, X) equals Q.
Let N ∈ Z≥3 . Let T be a Q-scheme. To T we attach a set MN (T ) of triples (A, λ, φ), which takes a while
to define. First, A is an Abelian scheme over T , of relative dimension g. Let A∨ = Pic0 (A/T ) be the dual
∼
Abelian scheme. Then λ : A → A∨ is an isomorphism whose fibre λt : At → A∨
t is a polarization for every
geometric point t of T ( [MF82, Chap. 6, §2]). Let (Z/N Z)2g
be
the
constant
group scheme attached to
T
2g
∨
(Z/N Z) over T . Write e for the Weil paring on A[N ] × A [N ] with values in µN,T . Via λ we identify
A[N ] with A∨ [N ], and we obtain the λ-Weil paring eλ : A[N ] ×T A[N ] → µN,T . A level structure is the
∼
∼
datum consisting of an isomorphism φ : (Z/N Z)2g
T → A[N ] and an isomorphism ν : (Z/N Z)T → µN,T of
T -schemes, such that the diagram
2g
(Z/N Z)2g
T ×T (Z/N Z)T
φ
h·,·istd
/ (Z/N Z)T
ν
A[N ] ×T A[N ]
eλ
/ µN,T
commutes. The couple (φ, ν) is called a symplectic isomorphism. Two triples (A, λ, φ), (A0 , λ0 , φ0 ) are
∼
equivalent if there exists a T -isomorphism f : A → A0 such that λ = f ∗ (λ0 ) and φ = f ∗ (φ0 ). By pull-back
functoriality, the set MN (T ) defines a contravariant functor from the category of Q-schemes to sets.
b be the subgroup of all g ∈ GSp2g (Z)
b which reduce to 1 ∈ GSp2g (Z/N Z). The
Let K(N ) ⊂ GSp2g (Z)
Shimura variety SK(N ) := ShK(N ) (G, X) represents the moduli problem MN . We write (AN , λN , φN ) for
the universal triple over SN .
Similarly, the Shimura variety S := Sh(G, X) represents the moduli problem M over Q which parametrizes
∼
b 2g →
triples (A, λ, φ), where φ : Z
T (A) is a symplectic trivialization of the Tate module. The universal
triple over S is (p∗N AN , p∗N λN , ϕ) with ϕ ⊗ Z/N = ϕN and where pN : S SK(N ) denotes the canonical
projection. In particular, the Tate module of the Abelian variety AN /SK(N ) becomes trivial when base
changed to S. Moreover, the Shimura variety S is universal for this property:
Lemma 5.2. For any triple (T, f, ϕ), where f : T → SK(N ) is a morphism of Q-schemes and ϕ :
∼ b 2g
T (f ∗ AN ) → Z
a sympletic trivialization such that ϕ ⊗ Z/N Z = f ∗ φN , there exists a unique morphism u : T → S, such that f = pN ◦ u and ϕ = u∗ φ.
Proof. We have (f ∗ AN , f ∗ λN , ϕ) ∈ M(T ). Since S represents M we obtain the required morphism
u : T → S. Fix a connected component SN of SK(N ) . Let s ∈ S, sN its image in SN and s, sN geometric points
above s and sN respectively. The fundamental group π1 (SN ; sN ) acts (on the left) on the fibre p−1
N (sN ).
This fibre is also a right K(N )-torsor, and the morphisms (·g) : S → S for g ∈ K(N ), are automorphisms
of the pro-étale cover pN : S SN . Hence the action of π1 (SN ; sN ) commutes with the K(N )-action.
Thus, using the choice of base point s ∈ p−1
N (sN ), we obtain a unique morphism ρ : π1 (SN ; sN ) → K(N )
such that for all σ ∈ π1 (SN ; sN ) we have σs = sρ(σ).
We can construct a second representation. We have the representation
(5.1)
π1 (SN ; sN ) −→ GSp(T (AsN ), ν −1 ◦ eλ )
on the Tate module of AsN (Paragraph 3.3.3). The point s ∈ p−1
N (sN ) induces a (symplectic) trivialization
∼ b 2g
φ : T (AsN ) → Z , and hence an isomorphism
(5.2)
∼
b
GSp(T (AsK ), ν −1 ◦ eλ ) −→ GSp2g (Z)
14
ANNA CADORET AND ARNO KRET
b The kernel U ⊂ π1 (SN , sN ) of
Thus, (5.1) and (5.2) induce a morphism ρA : π1 (SN , sN ) → GSp2g (Z).
the representation ρA corresponds, via Galois theory, to the pro-étale cover (SN )U of SN which trivializes
the Tate module of AN . Consequently, by Lemma 5.2 we have (SN )U = S and hence ρ = ρA .
5.2.2. Functoriality. For every morphism f : (G2 , X2 ) → (G1 , X1 ) of Shimura data and neat compact
open subgroups K1 ⊂ G1 (Af ), K2 ⊂ G2 (Af ) such that f (K2 ) ⊂ K1 , we get a commutative diagram of
Shimura varieties over E2 := E(G2 , X2 )
Sh(G2 , X2 )
p2
f
ShK2 (G2 , X2 )
/ Sh(G1 , X1 )E
2
f
p1
/ ShK (G1 , X1 )E
1
2
Fix a connected component X2+ ⊂ X2 mapping to a connected component X1+ ⊂ X1 . For i = 1, 2,
set Γi := Ki ∩ Gi (Q)+ , SΓi := ShcΓi (Gi , Xi+ ), EΓi := EΓi (Gi , Xi ). Fix si ∈ Shc(Gi , Xi+ ) lying over
si,Γi ∈ SΓi , i = 1, 2 such that f (s2 ) = s1 . Then the following diagram commutes
KO 2
f
/ K1
O
ρs2
π1 (SΓ2 , s2,Γ2 )
ρs1
f
/ π1 (SΓ E , s1,Γ ).
1 Γ2
1
5.3. Galois-generic and Hodge-generic points, comparison with Pink’s definition. In this section, we recall the definition of Hodge-generic point and show that `-Galois generic implies Hodge generic
(Lemma 5.3) . We also compare our definition of Galois-generic points with the one of Pink and show
that they coincide (Proposition 5.4).
For every x : S → GR in X, let M T (x) ⊂ G denote the Mumford-Tate group of x that is the smallest
algebraic subgroup M ⊂ G defined over Q and such x : S → GR factors through MR . Then, under Condition (4) G is the subgroup generated by the M T (x), x ∈ X and one says that x ∈ X is Hodge-generic
if M T (x) = G. The set of Hodge-generic points in analytically dense in X. Being Hodge-generic only
depends on the G(Q)-conjugacy class of x so we will say that s ∈ Sh(G, X) (resp. sK ∈ ShK (G, X))
is Hodge-generic if sC ∈ Sh(G, X)(C) = G(Q)\X × G(Af ) is the image of a point (x, g) with x ∈ X
Hodge-generic.
From the elementary properties (3) and (4) in Subsection 3.2, sK ∈ SΓ is Galois-generic with respect to
ρs : π1 (SΓ ; sK ) → K ⊂ G(Af ) if and only if sK 0 ∈ SΓ0 is Galois-generic with respect to ρs |π1 (SΓ0 ,sK 0 ) :
π1 (SΓ0 , sK 0 ) → K 0 ⊂ G(Af ). We will thus say that s ∈ Sh(G, X) is Galois-generic.
Lemma 5.3. `-Galois-generic points are Hodge-generic.
Proof. (See also [P05, Prop. 6.7]). Let s ∈ Sh(G, X) Galois-generic but such that M T (x) ( G.
Let x ∈ X lifting s and let Xx denote the M T (x)(R)-conjugacy class of x : S → M T (x)R . Then
the inclusion M T (x) ,→ G induces a closed immersion Sh(M T (x), Xx ) ,→ Sh(G, X) ( [D71, Prop.
1.15]). So for every neat compact open subgroup K ⊂ G(Af ), setting Kx := K ∩ M T (x)(Af ), the
induced morphism ShKx (M T (x), Xx ) → ShK (G, X) is the composite ofQa closed immersion followed by
an étale cover. Without loss of generality, we may assume that K = ` K` . Write Γ := K ∩ G(Q)+ ,
Γx := Kx ∩ M T (x)(Q)+ , let Xx+ ⊂ Xx , X + ⊂ X denote the connected component of x. Write sx ∈
SΓxx ,C := ShcΓx (M T (x), Xx+ )C for the image of x in ShcΓx (M T (x), Xx+ )C . By construction, the morphism
SΓxx ,C → SΓ,C maps sx to s and descends to a morphism of EΓx -schemes SΓxx → SΓ ×EΓ EΓx , where
EΓx := EΓx (M T (x), Xx+ ), EΓ := EΓ (G, X + ). By functoriality of the adelic representations we get the
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
commutative diagram
KO x 
/K
O
π1 (SΓxx )
O
/ π1 (SΓE )
O Γx
Galk(sx )
/ Galk(s)
15
.
As s ∈ SΓ is `-Galois-generic, this shows that Kx,` ⊂ M T (x)(Q` ) contains an open subgroup of Γ−
` . As
der
der
such a subgroup is Zariski-dense in GQ` , this shows that M T (x)Q` contains GQ` hence that M T (x) contains Gder . In particular, M T (x) is normal in G. In particular every G(R)-conjugates of x : S → M T (x)R
factors through M T (x)R , which contradicts the fact that G is the generic Mumford-Tate group of (G, X).
If we reformulate in terms of the adelic representation the notion of Galois-generic points as introduced
by Pink ( [P05, Def. 6.3]), we get that s ∈ Sh(G, X) is Galois-generic in the sense of Pink if and only if
sE ab ∈ Sh(G, X)E ab is Galois-generic (in our sense) that is sK,E ab ∈ SΓE ab is Galois-generic with respect
to ρs |π1 (SΓEab ) : π1 (SΓE ab ) → K ⊂ G(Af ) for one (or equivalently every) neat compact open subgroup
K ⊂ G(Af ). Here E ⊂ E ab ⊂ E denotes the maximal abelian extension of the reflex field E. Recall
that EΓ ⊂ E ab . With the notation of Subsection 3.2 (3), one has Ẽ ⊂ E ab but, in general the extension
Ẽ ⊂ E ab is not finite (when the reciprocity map describing the action of Galab
E on π0 (Sh(G, X)) has
infinite kernel). Still, the two notion of Galois-genericity coincide, namely
Proposition 5.4. A point s ∈ Sh(G, X) is Galois-generic if and only if sE ab ∈ Sh(G, X)E ab is Galoisgeneric.
Proof. This follows from Lemma 3.1, the fact that Π = Γ− ⊂ G(Af ) and that every open subgroup of Γ−
has finite abelianization. For this last assertion, observe that every open subgroup of Γ− is of the form
∆− for some arithmetic subgroup ∆ ⊂ Γ. Indeed, since Γ− is profinite, for every open subgroup U ⊂ Γ− ,
∆ := Γ ∩ U ⊂ Γ has finite index [Γ− : U ] (hence is an arithmetic subgroup) and satisfies ∆− = U . As a
result, the assertion follows from the group-theoretical Lemma 5.5 below. Lemma 5.5. Let G be a semisimple algebraic group over Q. For every arithmetic subgroup Γ ⊂ G(Q),
Γ− ⊂ G(Af ) has finite abelianization.
Proof. From the following commutative diagram
Γ
_ 
/ Γ− ,
=
Γ∧
where Γ ,→ Γ∧ denotes profinite completion, it is enough to show that Γ has finite abelianization.
From the Borel density theorem ( [Bo66a]), Γ is Zariski-dense in G. Fix a finite-dimensional faithful
Q-rational representation V of G and a Γ-invariant Z-lattice Λ ⊂ V . Fixing a Z-base of Λ we get an
embedding of G in GLn,Q such that Γ ⊂ G(Z) := G(Q) ∩ GLn (Z). For a prime `, let p` : GLn (Z) GLn (F` ) denote the reduction modulo-` morphism and set Γ` := p` (Γ). Then, from [N87, Thm. 5.1], we
have
G(F` )+ ⊂ Γ` ⊂ G(F` )
for ` 0 (depending only on n). Here G(F` )+ ⊂ G(F` ) denotes the subgroup generated by the order-`
elements in G(F` ) (or, equivalently, the `-Sylow subgroups as soon as ` > n). As G is semi simple,
G(F` )/G(F` )+ is abelian of order ≤ 2n−1 . In particular, there exists an integer N ≥ 1 such that Γ̃` :=
+
p−1
` (G(F` ) ) ⊂ Γ is a normal subgroup of index ≤ N . As Γ is an arithmetic subgroup, Γ is finitely
16
ANNA CADORET AND ARNO KRET
generated. In particular, it has only finitely many subgroups of index ≤ N . So Γ̃ := ∩` Γ̃` ⊂ Γ is again
a subgroup of finite index and, without loss of generality, we may replace Γ with Γ̃ that is assume that
Γ` = G(F` )+ for ` 0. As G is semisimple, G(F` ) hence Γ` = G(F` )+ acts semisimply on Fn` for ` 0.
ab
This implies that Γ` = 0 ( [CT14, Lemma 3.4]). Let Γ−
` ⊂ G(Z` ) denote the `-adic closure of Γ in
G(Q` ) (or, equivaletly, the `-adic component of Γ− ). As G is connected and semisimple, [C15, Remark
2.5] shows that the restriction of the modulo-` morphism
p` |Γ− : Γ−
` Γ`
`
is Frattini for ` 0. As
− −
[Γ−
` , Γ` ]
maps onto [Γ` , Γ` ] = Γ` , we obtain that
− −
−
[Γ−
` , Γ` ] = Γ`
+
for ` 0. The assumption that Γ` = Γ` also ensures that the product representation
Y
Y
Γ− →
Γ−
⊂
G(Q` )
`
`
`
is almost independant (see [CT14, §4.1,
that is, after
again Γ with a finite index
Q4.3]− for details)
Q replacing
−
− −
−
−
−
−
subgroup, we may assume that Γ = ` Γ` hence [Γ , Γ ] = ` [Γ` , Γ` ] . So it only remains to show
ab is finite for small `. But otherwise, Γ− would have an infinite abelian quotient Γ− Z . As
that (Γ−
`
` )
`
`
−
Γ` is a `-adic Lie group, so is Z` hence Z` has dimension ≥ 1. From exactness of the Lie-algebra functor
in the category of `-adic Lie groups, we obtain a surjective morphism of Lie algebras
Lie(Γ−
` ) Lie(Z` ).
But, on the other hand, since G is semisimple, we also have Lie(Γ−
` ) = Lie(G) ⊗ Q` , which has no abelian
quotient as a Lie algebra. 6. Conjecture 3.5 for adelic representations attached to connected Shimura data of
Abelian type
The main result of this section is Theorem 1.1, which asserts that Conjecture 3.5 holds for adelic representations attached to connected Shimura data of Abelian type. The idea of the proof consists in reducing
successively to the case of Shimura varieties of Hodge type and to Siegel Shimura varieties, for which
the result follows from the moduli interpretation and Conjecture 3.5 for Abelian schemes (Theorem 3.7).
While the reduction Hodge type ⇒ Siegel ⇒ Abelian scheme is standard, the reduction Abelian type ⇒
Hodge type is more delicate and, to handle it, one has to study transfer of Galois-genericity for Shimura
data whose adjoint Shimura data are identified by an isogeny of their derived subgroups (Subsection 6.2).
6.1. Shimura data of Hodge type. A Shimura datum (G2 , X2 ) (as well as the associated Shimura
variety) is said to be of Hodge type if there exists a morphism of Shimura data (G2 , X2 ) ,→ (G1 , X1 ) with
(G1 , X1 ) a Siegel Shimura datum and such that the underlying morphism of algebraic groups G2 → G1
is a closed immersion. The induced morphism of Shimura varieties Sh(G2 , X2 ) ,→ Sh(G1 , X1 ) is then a
closed immersion ( [D71, Prop. 1.15]).
Let (G2 , X2 ) be a shimura datum of Hodge type and fix an embedding f : (G2 , X2 ) ,→ (G1 , X1 ) of
Shimura data with (G1 , X1 ) a Siegel Shimura datum. Let K1 ⊂ G1 (Af ), K2 ⊂ G2 (Af ) be neat compact
open subgroups such that f (K2 ) ⊂ K1 . We use the same notation as in Subsection 5.2.2. Let A → SΓ1
denote the universal Abelian scheme over SΓ1 . Then the adelic representation
f
ρs
π1 (SΓ2 ) → π1 (SΓ1 EΓ2 ) →1 K1
coincides with the adelic representation attached to the Abelian scheme A|SΓ2 := A ×SΓ1 EΓ SΓ2 → SΓ2 .
2
But, as f : K1 ,→ K2 is injective, the commutativity of the second diagram in Paragraph 5.2.2 shows
f
ρs
that π1 (SΓ2 ) → π1 (SΓ1 EΓ2 ) →1 K1 and ρs2 : π1 (SΓ2 ) → K2 ,→ K1 have the same Galois-generic points.
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
17
To sum it up, we obtain
Lemma 6.1. Fix a prime `. With the above notation and assumptions, the following subsets of SΓ2
coincide
-
Galois-generic points with respect to ρs2 : π1 (SΓ2 ) → K2 ⊂ G(Af );
`-Galois-generic points with respect to ρs2 : π1 (SΓ2 ) → K2 ⊂ G(Af );
Galois-generic points with respect to the adelic representation attached to A|SΓ2 → SΓ2 ;
`-Galois-generic points with respect to the adelic representation attached to A|SΓ2 → SΓ2 .
6.2. Transfer of Galois-genericity. Let (Gi , Xi ), i = 1, 2 be Shimura data. Assume that there exists an
der
ad ad ˜
ad , X ad ).
isogeny φ : Gder
2 → G1 which induces an isomorphism of adjoint Shimura data φ : (G2 X2 )→(G
1
1
Let K1 ⊂ G1 (Af ), K2 ⊂ G2 (Af ) be neat compact open subgroups such that φ(K2 ∩ Gder
2 (Af )) ⊂ K1 .
der
Write Γ1 := K1 ∩ G1 (Q)+ ⊂ Gder
1 (Q), Γ2 := K2 ∩ G2 (Q)+ ⊂ G2 (Q) for the corresponding arithmetic
+
subgroups. Fix a connected component X ⊂ X2 which we identify with its image in X1 and X ad (recall
that X1 ⊂ X2 ⊂ X ad ). Write SΓi := ShcΓi (Gi , X + ), Ei := E(Gi , Xi ) and EΓi := EΓi (Gi , X + ), i = 1, 2.
Eventually, let F denote the compositum of EΓ1 and EΓ2 in Q. The étale cover φ : SΓ2 C → SΓ1 C descends
to an étale cover φ : SΓ2 F → SΓ1 F .
Lemma 6.2. (Transfer) For every si ∈ Shc(Gi , Xi+ ), i = 1, 2 such that φ(s2,K2 F ) = s1,K1 F , we have s2 ∈
Shc(G2 , X2+ ) is Galois-generic if and only if s1 ∈ Shc(G1 , X1+ ) is Galois-generic. The same statement
holds for `-Galois-generic points.
Proof. The proof for `-Galois-generic points is similar but easier so we only consider the case of Galoisgeneric points.A priori, we cannot compare directly ρ1 : π1 (SΓ1 F ) → K1 and ρ2 : π1 (SΓ2 F ) → K2 since
K1 and K2 are not directly related. But this becomes possible if one replaces F with F ab . Note that
Eiab ⊂ F ab is a finite extension, i = 1, 2. The following commutative diagram summarizes the situation
φ
O oo
Γ−
2 _ d
π1 (S Γ2 )
_

/ π1 (S Γ )
_ 1
π1 (SΓ2 F ab ) _
K2 o
ρ2
π1 (SΓ2 F ) 

/ /, Γ−
: 1 _
/ π1 (S
Γ1 F ab )
_
/ π1 (SΓ F )
1
ρ1
/ K1
Let s2 ∈ SΓ2 F mapping to s1 ∈ SΓ1 F . Consider the following assertions
(1)
(2)
(3)
(4)
s2 ∈ SΓ2 F is Galois-generic with
s2,F ab ∈ SΓ2 F ab is Galois-generic
s1,F ab ∈ SΓ2 F ab is Galois-generic
s1 ∈ SΓ2 F is Galois-generic with
respect to ρ2 : π1 (SΓ2 F ) → K2 ⊂ G2 (Af );
with respect to ρ2 : π1 (SΓ2 F ab ) → Γ−
2 ⊂ G2 (Af );
with respect to ρ1 : π1 (SΓ1 F ab ) → Γ−
1 ⊂ G1 (Af );
respect to ρ1 : π1 (SΓ1 F ) → K1 ⊂ G1 (Af ).
Then we have (1) ⇔ (2) ⇒ (3) ⇔ (4). The equivalences (1) ⇔ (2) and (3) ⇔ (4) follow from Corollary
−
5.4 while the equivalence (2) ⇔ (3) comes from the fact that the induced morphism φ : Γ−
2 → Γ1 has
open image (note that it is not injective in general).
The converse implication (3) ⇒ (2) is more delicate.
Γ−
2
Indeed, (3) implies that Πs2,K
Γ−
1.
2 ,F
ab
:= ρ2 ◦
σs2 ,K2 (Galk(s2 )F ab ) ⊂
maps onto an open subgroup of
From this, we would like to deduce
−
that Πs2,K ,F ab ⊂ Γ2 is an open subgroup. This follows from the group-theoretical Lemma 6.3 below. 2
Lemma 6.3. Let φ : G2 → G1 be an isogeny of connected semisimple algebraic groups over Q. For
i = 1, 2, let Γi ⊂ Gi (Q) be an arithmetic subgroup and let Γ−
i ⊂ Gi (Af ) denote its adelic closure. Assume
−
−
that φ(Γ2 ) ⊂ Γ1 . Then for every closed subgroup ∆ ⊂ Γ2 , φ(∆) ⊂ Γ−
1 is open if and only if φ(∆) ⊂ Γ2
is open.
18
ANNA CADORET AND ARNO KRET
Proof. The if implication is straightforward. We prove the only if one. We retain the notation of the
proof of Lemma 5.5. We may replace Γ1 , Γ2 by smaller
In particular, we may
Q − arithmetic subgroups.
−
+
assume (see the proof of Theorem 5.5) that Γi = ` Γi,` , Γi,` = Gi (F` ) and that the restriction of the
reduction-modulo-` morphism p` |Γ−,` : Γ−
i,` Γi,` is Frattini for ` 0 and i = 1, 2. Furthermore, for
i
` 0 the isogeny φ : G2 → G1 induces a Frattini cover5
+
φ` : G2 (F` )+ Gder
1 (F` ) .
Let ∆` ⊂ Γ1,` denote the `-adic component of ∆. The following diagram sums up the situation.
Γ1,` = G2 (F` )+
φ`
OO
p`
/ / G1 (F` )+ = Γ2,`
OO
p`
∆` ⊂ Γ−
1,`
φ
/ / Γ−
2,`
−
As φ(∆) ⊂ Γ−
1 is open, ∆` surjects onto Γ1,` for ` 0. So, as φ` : Γ2,` Γ1,` is Frattini, ∆` also surjects
onto Γ2,` . But as p` |Γ− : Γ−
Γ2,` is Frattini, ∆` = Γ−
2,` . In particular, there exists an open subgroup
2,` Q 2,`
0
0
0
∆ ⊂ ∆ such that ∆ = ` ∆` (see [CT14, §4.1, 4.3] for details). The conclusion follows from the facts
Q −
−
−
0
0
that Γ−
2 =
` Γ2,` , ∆` = ∆` = Γ2,` for ` 0 and ∆` ⊂ ∆` ⊂ Γ2,` are open subgroups for every `. 6.3. Shimura data of Abelian type. A Shimura datum (G1 , X1 ) (as well as the associated Shimura
variety) is said to be of Abelian type if there exists a Shimura datum (G2 , X2 ) of Hodge type and an isogeny
der which induces an isomorphism of adjoint Shimura data φ : (Gad , X ad )→(G
ad
φ : Gder
˜ ad
2 → G1
2
2
1 , X1 ).
End of the proof of Theorem 1.1: We retain the notation of Subsection 6.2 and we apply Lemma
6.2 twice. Start from a `-Galois-generic point s1 ∈ Sh(G1 , X1 ) and fix s2 ∈ Sh(G2 , X2 ) such that
φ(s2,K2 ,F ) = s1,K1 ,F . As s1 ∈ Sh(G1 , X1 ) is `-Galois-generic, from the if implication in Lemma 6.2,
s2 ∈ Sh(G2 , X2 ) is `-Galois-generic. As (G2 , X2 ) is of Hodge-type, It follows from Lemma 6.1 that
s2 ∈ Sh(G2 , X2 ) is Galois-generic. Thus, from the only if implication in Lemma 6.2, s1 ∈ Sh(G1 , X1 ) is
Galois-generic. We refer to [D71, §1.2, 1.3 and 2.7] and [Mi13, §10] for a detailled account of Shimura data of Abelian
type. These include essentially all Shimura data (G, X) except those for which G has simple factors of
type E6 , E7 and mixed type DR × DH .
7. Equidistribution
The aim of this Section is to prove Theorem 1.2, which extends Pink’s equidistribution theorem for
Galois-generic points from Siegel Shimura varieties to arbitrary Shimura varieties. Before turning to the
proof itself, we review the definition of generalized Hecke operators.
7.1. Generalized Hecke operators. For every a ∈ G(Af ), let Ta denote the corresponding Hecke
operator, that is, the algebraic correspondence
·a−1
ShK (G, X)C ← Sh(G, X)C →
˜ Sh(G, X)C → ShK (G, X)C.
It is part of the definition of a canonical model that Ta is defined over E(G, X). For a ∈ G(Q)+ K, Ta
restricts to an algebraic correspondence (of degree 1 if a ∈ K) defined over E(Γ∩a−1 Γa) (G, X + )
a·
ShcΓ (G, X + )C ← Shc(G, X + ))C →
˜ Shc(G, X + )C → ShcΓ (G, X + )C .
5More generally, if psc : Gsc → G is the simly connected cover of a semisimple group over F and ` > | ker(psc )| then the
`
induced morphism Gsc (F` ) → G(F` )+ is a Frattini cover. Indeed, let H ⊂ Gsc (F` ) mapping surjectively onto G(F` )+ . Let
Z := ker(psc ). Then HZ = Gsc (F` ) hence H is normal in Gsc (F` ). This yields a quotient Gsc (F` ) Gsc (Fp )/H which is of
prime-to-`. As Gsc (F` ) = Gsc (F` )+ , this forces H = Gsc (F` ).
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
19
For s = Γx ∈ ShcΓ (G, X + )(C) we set
Ta (s) = {Γaγx | γ ∈ Γ}.
For a subset A ⊂ G(Q)+ and s ∈ ShcΓ (G, X + )C , we also set
[
ΘA (s) :=
Ta (s) ⊂ ShΓ (G, X + )(C)
a∈A
(or ΘΓ,A (s)) for its A-Hecke orbit. For A = G(Q)+ we simply write ΘA (s) =: Θ(s) (or ΘΓ (s)) and call it
the Hecke orbit of s.
Let Aut(G, X + ) denote the group automorphisms of G defined over Q and stabilizing X + . For every
φ ∈ Aut(G, X + ), the corresponding generalized Hecke operator Tφ is the algebraic correspondence
φ·
ShcΓ (G, X + )C ← Shc(G, X + ))C →
˜ Shc(G, X + )C → ShcΓ (G, X + )C .
For s = Γx ∈ ShcΓ (G, X + )(C) we set
Tφ (s) = {Γφ(γ)φ(x) | γ ∈ Γ}.
For a subset Φ ⊂ Aut(G, X + ) and s ∈ ShΓ (Gder , X + )C , we also write
[
ΘΦ (s) :=
Tφ (s) ⊂ ShΓ (Gder , X + )(C)
φ∈Φ
(or ΘΓ,Φ (s)) for its Φ-Hecke orbit. The usual Hecke orbit Θ(s) coincides with the Φ-Hecke orbit ΘΦ (s)
with Φ the image of G(Q)+ → Aut(G, X + ) given by inner automorphisms. For A = Aut(G, X + ) we
b
b Γ (s)) and call it the generalized Hecke orbit of s.
simply write ΘΦ (s) =: Θ(s)
(or Θ
For the comparison between usual Hecke orbits and generalized Hecke orbits, see [O13, §4.1.1]; let us only
point out the following observation, which will be used in the proof of Theorem 1.2.
Let (G, X + ) be a connected Shimura datum with G adjoint. For an arithmetic subgroup Γ ⊂ G(Q)+ ,
write SΓ := ShcΓ (G, X + ) and for arithmetic subgroups Γ0 ⊂ Γ, write pΓ0 ,Γ : SΓ0 → SΓ for the canonical
projection. Then,
Lemma 7.1. For every arithmetic subgroup Γ ⊂ G(Q)+ there exists an arithmetic subgroup Γ0 ⊂ Γ such
that, for every subset Φ ⊂ Aut(G, X + ) and s ∈ SΓ , p−1
Γ0 ,Γ (ΘΦ (s)) is contained in a finite union of usual
Hecke orbits on SΓ0 .
Proof. As G is adjoint, the quotient Aut(G, X + )/G(Q)+ is finite. Choose a system of representives
φ1 , . . . , φr for Aut(G, X + )/G(Q)+ and set
\
\
Γ0 :=
φi (Γ) ⊂ Γ, Γ00 :=
φi (Γ0 ) ⊂ Γ0 .
1≤i≤r
1≤i≤r
Then Γ00 , Γ0 ⊂ Γ are again arithmetic subgroups. Fix systems of representatives γj , j = 1, . . . s of Γ/Γ0 ,
αk , k = 1, . . . t of Γ/Γ00 and αi,l , l = 1, . . . , ti of φi (Γ0 )/Γ00 for i = 1, . . . , r. For an arbitrary element
φ = inn(a) ◦ φi ∈ Aut(G, X + ), we can then compute explicitly
[ [ [
p−1
Tαk aαl,i ,Γ00 (φi (γj x)).
Γ00 ,Γ (Tφ,Γ (s)) =
1≤j≤s 1≤k≤t 1≤l≤ti
This shows that for every subset Φ ⊂ Aut(G, X + ) and s ∈ SΓ , p−1
Γ00 ,Γ (ΘΦ (s)) is contained in a finite union
of usual Hecke orbits on SΓ00 . 20
ANNA CADORET AND ARNO KRET
7.2. Equidistribution. Let G be a connected Q-simple algebraic group of non-compact type and let
G(R)+ ⊂ G(R) denote the connected component of 1 in G(R). Set G(Q)+ := G(Q) ∩ G(R)+ . Fix an
arithmetic subgroup Γ ⊂ G(Q)+ . Then Γ is a lattice in G(R)+ ( [BoH62]); let µ denote the normalized
Haar measure on Γ \ G(R)+ . For a function f : Γ \ G(R)+ → C and an element a ∈ G(Q)+ define its
Hecke transform Ta (f ) : Γ \ G(R)+ → C by
X
1
f (y 0 ).
Ta (f )(y) =
degΓ (a) 0
y ∈Ta (y)
We then have the following equidistribution result for Hecke orbits.
Fact 7.2. (Equidistribution - [EO06, Thm. 1.2]; see also [BuS91, ‘Thm. 5.2’]) For every sequence
a = (an ) of elements in G(Q)+ with lim degΓ (an ) = +∞ and for every continuous bounded function
n→+∞
f : Γ \ G(R)+ → C and y ∈ Γ\G(R)+ we have
Z
lim Tan (f )(y) =
n→+∞
f dµ.
Γ\G(R)+
In particular, for every y ∈ Γ\G(R)+ the set
Θa (y) :=
[
Tan (y)
n≥1
is dense in Γ \ G(R)+ .
7.3. End of the proof. We follow Pink’s strategy, replacing the results of [ClOU01] with Fact 7.2. To
apply Fact 7.2, we need the following general finiteness result about Hecke operators of bounded degree.
Theorem 7.3. Let G be a connected semisimple group over Q of non-compact type and let Γ ⊂ G(Q)
be an arithmetic subgroup. Then, for every integer d ≥ 1 there are only finitely many double-classes
ΓaΓ ∈ Γ \ G(Q)/Γ with |Γ \ ΓaΓ| ≤ d.
We refer to the appendix for a proof of this result.
Let (G, X) be a Shimura datum, X + ⊂ X a connected component, K ⊂ G(Af ) a neat compact open
subgroup. Write Γ := K ∩ G(Q) ⊂ Gder (Q)+ and SΓ := ShcΓ (G, X + ).
Corollary 7.4. (Compare with [P05, Thm. 7.5]) For every s ∈ SΓ and for every sequence φ = (φn ) of
elements in Aut(G, X + ), the set Θφ (s) is either finite or Zariski-dense in SΓ .
Proof. The morphism pad G → Gad induces an isomorphism of schemes over C
pad : SΓC → SΓad C := ShcΓad (Gad , X + )C
which maps Θφ,Γ (s) bijectively onto Θπ(φ),Γad (pad (s)). Thus, without loss of generality, we may replace
G by Gad and Γ by Γad that is, assume that G is adjoint. Next, for every arithmetic subgroup Γ0 ⊂ Γ,
the quotient map pΓ0 ,Γ : SΓ0 C → SΓC is a finite cover. So we may replace Γ by Γ0 and Θφ,Γ (s) with
p−1
Γ0 ,Γ (Θφ,Γ (s)). In particular (Lemma 7.1), we may assume that φn = an ∈ G(Q)+ , n ≥ 0. Then, consider
zar
the analytic map p : Γ\G(R)+ SΓ (C), Γg → g · x. Assume that Θa (s)
=: Z ( SΓ is a strict closed
subscheme. In particular, Z(C) ( SΓ (C) is a strict closed analytic subset. As p : Γ\G(R)+ SΓ (C) is
analytic and surjective, p−1 (Z(C)) ( Γ\G(R)+ is again a strict closed analytic subset. As p−1 (Θa (s)) =
Θa (1) ⊂ p−1 (Z(C)), we have Θa (1)− ⊂ p−1 (Z(C))− = p−1 (Z(C)) ( Γ\G(R)+ . Theorem 7.2 thus implies
that degΓ (an ) is bounded and, by Theorem 7.3, this in turn ensures that the set
{Γan Γ | n ≥ 0} ⊂ Γ\G(Q)/Γ
is finite hence a fortiori that Θa (s) = p(Θa (1)) is. End of the proof of Theorem 1.2: (Compare with [P05, Proof of Thm. 7.6]). As usual, write
E := E(G, X) and EΓ := EΓ (G, X + ).
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
21
b Γ (s).
Let Z ,→ SΓC be a closed subvariety containing an infinite subset of Θ
- Reduction to the case where s ∈ SΓ is strictly Galois-generic and Z is defined over the
residue field k(s) of s: As SΓ is of finite type over EΓ , Z is defined over a finitely generated extension
F of EΓ and sF ∈ SΓF is again Galois-generic. In particular, we can find a congruence subgroup Γ0 ⊂ Γ
0
such that Γ − ⊂ ΠsF . Hence, up to replacing F with F EΓ0 k(s), we may assume that sF lifts to a
strictly Galois-generic point s0F ∈ SΓ0 F (Subsection 3.1, Property (5)). Let p : SΓ0 F → SΓF denote the
b Γ0 (s0 )
quotient map. Then p−1 Z ,→ SΓ0 F is again a closed subvariety containing an infinite subset of Θ
F
(just observe that
[
p−1 (Tφ,Γ (s)) =
Tinn(γ1 (φ(γ2 )))φ,Γ0 ((s0F ))
γ1 ,γ2 ∈Γ/Γ0
for every φ ∈
As it is enough to show that p−1 Z = SΓ0 F is Zariski-dense, up to replacing
EΓ with F , SΓ with SΓ0 F , s with s0F and Z with p−1 Z without loss of generality, we may assume that
s ∈ SΓ is strictly Galois-generic and that Z is defined over k(s).
Aut(G, X + )).
- Conclusion: As s is strictly Galois-generic, for every φ ∈ Aut(G, X + ), the group Galk(s) acts transitively on the inverse image of s in any connected component S̃K∩φ−1 (K) of
SK∩φ−1 (K) := ShK∩φ−1 (K) (G, X) ×ShK (G,X) SΓ .
As GalEΓ∩φ−1 (Γ) ⊂ GalEΓ is the stabilizer of SΓ∩φ−1 (Γ) ⊂ S̃K∩φ−1 (K) ×EΓ EΓ∩g−1 Γg , the subgroup
GalEΓ∩φ−1 (Γ) k(s) ⊂ Galk(s) acts transitively on the inverse image of s in SΓ∩φ−1 (Γ) hence on Tφ (s) ⊂ SΓ .
In particular, for every φ ∈ Aut(G, X + ) such that Z ∩ Tφ (s) 6= ∅ we have Tφ (s) ⊂ Z (recall that we
assume Z is defined over k(s)). Now, consider a sequence φ = (φn ) of elements in Aut(G, X + ) such
that Z contains a point sn ∈ Tφn (s), n ≥ 1 with sn 6= sm for n 6= m. Then
[
Θφ (s) =
Tφn (s) ⊂ Z
n≥1
is infinite by construction. So, the conclusion follows from Corollary 7.4.
8. Appendix: Degree of Hecke operators
The aim of this appendix is to prove the following group-theoretical result.
Theorem 7.3. Let G be a connected semisimple group over Q of non-compact type and let Γ ⊂ G(Q)
be an arithmetic subgroup. Then, for every integer d ≥ 1 there are only finitely many double-classes
ΓaΓ ∈ Γ \ G(Q)/Γ with |Γ \ ΓaΓ| ≤ d.
Remark 8.1. (1) The assumption that G is of non-compact type is there to ensure that we have strong
approximation when G is simply connected. If we remove this assumption, Proposition 7.3 becomes
either trivially true or trivially false. Indeed, if G is Q-simple of compact type then Γ is always finite.
Hence either G(Q) is finite and Theorem 7.3 holds (e.g. G = SL1,D with D a quaternion algebra
over Q which is non-split at infinity) or G(Q) is infinite and Theorem 7.3 does not hold (e.g. G is
the totally definite orthogonal groups O(n) with n > 3).
(2) Actually, we deduce Theorem 7.3 from the a priori stronger adelic result of Proposition 8.5.
After a first version of this draft was released, Hee Oh mentionned to us that Theorem 7.3 could be
proved by ergodic technics. Here is the argument she explained to us, using the notation of [EO06,
Thm.1.2]. If we have infinitely many distinct Γai Γ whose degree is bounded by d, then the associated
∆(G)-invariant measures ν̃ai have a weak-limit nu.
˜ There are two possibilities for ν̃ as discussed in the
proof: either ν̃ is supported in the closed ∆(G)-invariant measure or is a G × G invariant measure. In
the first case, the proof shows that the sequence [(e, ai )]∆(G) has a constant sub-sequence; or equivalently, that the sequence Γai Γ has a constant subsequence, contradicting the assumption that they are
distinct. The second case where ν̃ is G×G-invariant cannot happen; since this is equivalent to Γ\Γai Γ
22
ANNA CADORET AND ARNO KRET
is equidistributed in Γ\G(R); but Γ\Γai Γ has at most d points, so cannot possibly be equidistributed.
We do not know whether our adelic statement (Proposition 8.5) can be recovered from Theorem 7.3
hence be proved by ergodic technics. In any case, our proof relies on different arguments (Bruhat-Tits,
strong approximation in particular) and, with some efforts, could be made effective.
In Subsection 8.1, we fix some notation and prove an elementary formal lemma (Lemma 8.4), which we
will use repeatedly in the following. The proof of the adelic variant of Theorem 7.3 (Proposition 8.5) is
carried out in Subsection 8.2; here the ‘natural’ tool are avatars of the Bruhat-Tits decomposition, which
provide explicit estimates for the local degrees even in the ramified case (see Subsection 8.2 for details).
Subsection 8.3, is devoted to the reduction from the global case to the adelic case and the conclusion
of the proof of Theorem 7.3. It uses strong approximation and is easy when G is assumed to be simply
connected but more technical in general.
8.1. Formal lemmas. We say that two subgroups K, K 0 ⊂ G are commensurable if K ∩ K 0 is of
finite index in both K and K 0 . Commensurability is an equivalence relation, which we denote by ≡,
on the set of subgroups of G. For K ⊂ G a subgroup, let K ≡ be the set of all g ∈ G such that
[K : K ∩ gKg −1 ] = |K\KgK| is finite. Then K ≡G ⊂ G is a subgroup containing K and for every
0
subgroup K 0 ⊂ G commensurable with K, we have K ≡G = K ≡G . We can thus define the degree of
g ∈ K ≡ with respect to K to be the positive integer
degK (g) := [K : K ∩ gKg −1 ] = |K\KgK| ∈ Z≥1 .
For K, K 0 ⊂ G commensurable subgroups, define the index of K 0 with respect to K to be the rational
number
[K : K ∩ K 0 ]
∈ Q>0 .
[K : K 0 ] :=
[K 0 : K ∩ K 0 ]
For subgroups K ⊂ U ⊂ G, let CorU (K) denote the largest subgroup of K which is normalized by U .
Explicitly,
CorU (K) :=
\
uKu−1 ⊂ K.
u∈U
Equivalently CorU (K) is the kernel of U acting on U/K by left-translation. In particular, if [U : K] is
finite then [U : CorU (K)] is also finite and [U : CorU (K)] ≤ [U : K]!.
0
Lemma 8.2. Let K, K 0 ⊂ G be two commensurable subgroups and let a ∈ K ≡ (= K ≡ ). Then we have
degK (a) ≤ CK,K 0 degK 0 (a),
where CK,K 0 := [K : CorK (K ∩
K 0 )][K 0
: CorK (K ∩ K 0 )].
Proof. Observe first that (i) if K 0 ⊂ K then degK 0 (a) ≤ [K : K 0 ] degK (a) and that (ii) if furthermore
K 0 ⊂ K is normal, then degK (a) ≤ [K : K 0 ] degK 0 (a). The proof of (i) is straightforward. As for the
proof of (ii), if K 0 ⊂ K is normal then
(8.1)
degK 0 (ka) = |K 0 /K 0 ∩ kaK 0 (ka)−1 | = |K 0 /k −1 K 0 k ∩ aK 0 a−1 | = degK 0 (a)
and similarly, degK 0 (ak) = degK 0 (a). Let R ⊂ K be a set of representatives for the left-cosets of K 0 in
K. By normality, R is a also a set of representatives for the right-cosets of K 0 in K. Hence,
0
[K : K 0 ] degK (a) = |K 0 \K||K\KaK| = |K
X\KaK|
X
≤
|K 0 \K 0 xayK 0 | =
deg(xay) = [K : K 0 ]2 degK 0 (a),
x,y∈R
x,y∈R
where the last equality follows from Equation (8.1). The assertion in Lemma 8.2 now follows from the
combination of (i) and (ii):
degK (a) ≤ [K : CorK (K ∩ K 0 )] degCorK (K∩K 0 ) (a)
(by (i))
0
0
0
≤ [K : CorK (K ∩ K )][K : CorK (K ∩ K )] degK 0 (a) (by (ii)).
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
23
Definition 8.3. For a subgroup K ⊂ G, we say that property ?(G, K) holds if for every integer d ≥ 1
there are only finitely many K-double classes KaK ∈ K\G/K with degK (a) ≤ d.
Lemma 8.4. Let ϕ : G0 → G be a morphism of groups and K 0 ⊂ G0 , K ⊂ G two subgroups. Assume that
ker(ϕ), [G : ϕ(G0 )] are finite and K, ϕ(K 0 ) ⊂ G are commensurable. Then ?(G, K) holds if and only if
?(G0 , K 0 ) holds.
Proof. First, consider the case where G0 = G and ϕ is the identity. As the situation is symmetric in
K, K 0 , it is enough to show the implication ?(G, K) ⇒ ?(G0 , K 0 ). Set K 00 := K ∩ K 0 and fix an integer
0
d ≥ 1. Let a ∈ K ≡ (= K ≡ ) with degK 0 (a) ≤ d. From Lemma 8.2.(iii), degK (a) ≤ CK 0 ,K d. From ?(G, K),
there are only finitely many possibilities for the K-double class KaK ∈ K \ G/K. But, then, there are
also only finitely many possibilities for the K 0 -double class K 0 aK 0 ∈ K 0 \ G/K 0 since the induced maps
K 00 \G/K 00 → K\G/K, K 00 \G/K 00 → K 0 \G/K 0 are both surjective with finite fibers.
In particular, ?(G, K) holds if and only if ?(G, ϕ(K 0 )) holds. So, in the following, we may and will assume
that K = ϕ(K 0 ).
Then the assumption that ker(ϕ) is finite ensures that the induced map K 0 \G0 /K 0 → K\G/K has finite
fibers.
The implication ?(G, K) ⇒ ?(G0 , K 0 ) then follows from
degK (ϕ(a0 )) ≤ degK 0 (a0 ), a0 ∈ G0 .
For the implication ?(G0 , K 0 ) ⇒ ?(G, K), observe that
degK (ab) ≤ degK (a) degK (b), a, b ∈ G
(just note that KabK ⊂ KaKbK). Let ∆ denote a (finite) set of representatives of left-cosets of ϕ(G0 )
in G. Then for every a ∈ G there exists (a unique) δa ∈ ∆ such that aδa−1 = ϕ(a0 ) for some a0 ∈ G0 . In
particular
degK 0 (a0 ) ≤ degK (ϕ(a0 )) ≤ min{degK (δ −1 ) | δ ∈ ∆} degK (a).
In particular, to prove Theorem 7.3, we may assume that Γ ⊂ G(Q) is a congruence subgroup. So, let
K ⊂ G(Af ) a compact open subgroup such that Γ = K ∩ G(Q). By shrinking K, we may assume it is of
the form
(8.2)
K = KP K P ,
where P is a finite set of primes containing the primes where G ramifies, KP =
Q
compact open and K P = p∈P
/ Kp with Kp ⊂ G(Qp ) hyperspecial.
Q
p∈P
Kp with Kp ⊂ G(Qp )
8.2. The local and Adelic cases. The aim of this section is to prove the adelic variant of Theorem
7.3.
Proposition 8.5. Let G be a connected semisimple group over Q then ?(G(Af ), K) holds for every
compact open subgroup K ⊂ G(Af ).
The proof of Proposition 8.5 relies on Lemmas 8.6 and Lemma 8.8 below for the degree of local Hecke
operators.
Lemma 8.6. Let G be a connected semisimple group over Qp then ?(G(Qp ), K) holds for every compact
open subgroup K ⊂ G(Qp ).
Lemma 8.7. Let G0 → G be an isogeny of algebraic groups over Qp . Assume that the degree N of its
kernel µ is at most p. Then there exists a constant ι(N ) depending only on N (but not on p) such that
|coker(G0 (Qp ) → G(Qp ))| ≤ ι(N ).
24
ANNA CADORET AND ARNO KRET
Proof. By the long exact sequence in Galois cohomology, it is enough to show that there exists a constant
ι(N ) depending only on N such that |H1 (Qp , µ(Qp ))| ≤ ι(N ). For this, let E/Qp denote the finite Galois
extension corresponding to the kernel of the action of GalQp on µ(Qp ). We have [E : Qp ] ≤ N !. Write
c(N ) :=
N
!N
X
Xn
n=1 d|n
d
for the number of extension of degree ≤ N !N of Qp (here, we use that p is prime-to-N !) and let EN /E
denote the finite Galois extension corresponding to the open subgroup
\
GalE 0 ⊂ GalE ,
E0
where E 0 /E describes all Galois extensions of degree ≤ N in Qp /E. We have [EN : Qp ] ≤ N !c(N )N . By
construction, the restriction map
H1 (Qp , µ(Qp )) → H1 (EN , µ(Qp ))
is trivial. So, by the inflation-restriction exact sequence, we get an isomorphism
˜ 1 (Qp , µ(Qp )).
H1 (EN , µ(Qp ))→H
So we can take ι(N ) := N N !c(N )N .
In the following, given a connected semi simple group G over a field k we let µG denote the kernel of the
simply connected cover G0 → G.
Lemma 8.8. Let G be a connected semisimple group over Qp and let K ⊂ G(Qp ) be a maximal special
compact subgroup. Assume that p > ι(|µG |). Then, for every a ∈ G(Qp ), either a ∈ K or degK (a) ≥ p.
Lemma 8.8 applies in particular when G is unramified and K ⊂ G(Qp ) is hyperspecial, which is the only
case we need.
Proof of Proposition 8.5. Recall
Q that we may assume that K is of the form (8.2). Then, for every
/ P and p > max{d, ι(|µG |)},
a ∈ G(Af ) we have degK (a) = p degKp (a). Assume degK (a) ≤ d. If p ∈
Lemma 8.8 shows that a ∈ Kp . But, then, the conclusion follows from Lemma 8.6 applied to the finitely
many p which are in P or ≤ max{d, ι(|µG |)}. The remaining part of this section is devoted to the proof of Lemma 8.6 and Lemma 8.8. Recall that for
an anisotropic group G over Qp the group G(Qp ) is compact ( [Pr82]). So it is enough to prove Lemma
8.6 and Lemma 8.8 for isotropic groups G. In that case, they follow from the Bruhat-Tits decomposition
attached to the euclidean Bruhat-Tits building of G(Qp ), which expresses explicitly the degree of local
Hecke operators in terms of the extended affine Weyl group of the building. For Lemma 8.6, we can use
Lemma 8.4 to reduce to the case where G is simply connected. Then the parametrizing group of the
Bruhat-Tits decomposition is a Coxeter group (the affine Weyl group) and computations are easy. For
Lemma 8.8, we can no longer resort to Lemma 8.4 and thus have to handle the Bruhat-Tits decomposition
for possibly non-simply connected G. There, the parametrizing group of the Bruhat-Tits decomposition
(the extended affine Weyl group) is a semi-direct product of a Coxeter group by a finite group of nonpreserving type automorphisms, which make computations slightly more technical. As this is possibly
less classically known to non-experts, we have included an expository paragraph, which we tried and keep
as self-contained as possible.
8.2.1. Review of Bruhat-Tits theory. Let G be a connected semisimple isotropic group over Qp with
Qp -split maximal torus S. The principle of Bruhat-Tits theory is to attach to G(Qp ) a discrete euclidean building endowed with a strongly transitive action of G(Qp ). The existence of this building is
essentially equivalent to the datum of a generalized Tits system (G(Qp ), B, N ) and once the existence
of (G(Qp ), B, N ) is established, the axiomatic of Tits systems gives a combinatorial description of the
compact subgroups of G(Qp ) containing B in terms of the extended affine Weyl group N/N ∩ B.
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
25
We assume the reader is familiar with the formalism of Tits systems and buildings ( [Bou68, Chap
IV], [BruT72, §1, §2], [Br89], [G13] etc.). We review below the description of the euclidean Bruhat-Tits
building attached to G(Qp ) and summarize the consequences of this construction we will need in Subsection 8.2.1.6. We follow closely the survey [R09], which provides a synthetic introduction to the classical
but less accessible [T79], [BruT72], [BruT84a], [BruT84b], [L95].
Given a group H acting on a set X and an element x ∈ X, we write Hx for the stabilizer of x in G.
8.2.1.1. The vectorial part of the extended affine Weyl group W v . Let X ∗ (S) and X∗ (S) denote the groups
of characters and cocharacters of S respectively. These are free Z-modules of rank r = dim(S), dual to
each other via the evaluation pairing X ∗ (S) × X∗ (S) → Z. Set V (S) := X∗ (S) ⊗Z R.
Set
N := NorG (S)(Qp ), Z := CenG (S)(Qp ).
The action of N on S by conjugation yields
ν v : N N/Z ,→ AutGrAlg/Qp (S) ,→ AutZ (X∗ (S)) ,→ GL(V (S)).
The torus S acts on the Lie algebra g of G, which decomposes accordingly as
M
g=
gα ,
α∈X ∗ (S)
where
gα :=
\
ker(s − α(s)Id).
s∈S
Let Φ denote the set of all 0 6= α ∈ X ∗ (S) such that gα 6= 0. Then Φ ⊂ V (S)∗ is a (not necessarily
reduced) root system in the usual sense. Let W v denote the Weyl group of Φ; we endow V (S) with
a W v -invariant scalar product. For every α ∈ Φ, let rα ∈ W v denote the orthogonal reflexion fixing
ker(α). The morphism ν v : N → GL(V (S)) induces an isomorphism ν v : N/Z →W
˜ v . We can describe
more precisely elements corresponding to the reflexions rα , α ∈ Φ. For every α ∈ Φ there exists a unique
connected unipotent group Uα ,→ G normalized by CenG (S) and such that the induced embedding of
Lie algebras uα ,→ g identifies uα with gα ⊕ g2α . Also, for every 1 6= u ∈ Uα (Qp ), the intersection
U−α (Qp )uU−α (Qp ) ∩ N consists of a single element m(u) which, furthermore, has the properties that
ν v (m(u)) = rα ∈ N/Z ' W v [BoT76, §5].
c . As the restriction map X ∗ (CenG (S)) → X ∗ (S) has finite
8.2.1.2. The extended affine Weyl group W
cockernel of order say m ≥ 1, for every z ∈ Z there exists a unique ν(z) ∈ V (S) such that χ(ν(z)) =
1
vp (mχ(z)), χ ∈ X ∗ (S). This defines a morphism ν : Z → V (S) characterized by the fact that
−m
χ(ν(s)) = −vp (χ(s)), χ ∈ X ∗ (S), s ∈ S(Qp ). Set Tb := ν(Z) ⊂ V (S) and Z0 := ker(ν) ⊂ Z, which is a
r
compact open subgroup containing S(Zp ) ' (Z×
p ) . The morphism ν : Z → V (S) extends [R09, Prop.
11.3] to a morphism
ν : N → GA(V (S))
which makes the following diagram commute
1
/Z
1
ν
/ V (S)
/N
ν
/ GA(V (S))
/ N/Z
/1
νv
/ GL(V (S))
/ 1,
where the left and right vertical arrows are the morphisms defined above. The extension ν : N →
c := ν(N ).
GA(V (S)) is unique up to GA(V (S))-conjugacy. Write W
26
ANNA CADORET AND ARNO KRET
8.2.1.3. The Weyl group W and the standard appartment A(S). For every α ∈ Φ and 1 6= u ∈ Uα (Qp ),
c is an orthogonal reflexion with hyperplane H(u) of direction ker(α) and
ν(m(u)) ∈ W
c
W := hν(m(u)) | 1 6= u ∈ Uα (Qp ), α ∈ Φi C W
is a discrete affine reflexion group. Let A(S) := (V (S), W ) denote the corresponding apartment (See
[BruT72, §1.3], [R09, Part I]) and F(S) the set of its facets. For a special point x ∈ A(S), we have
cx ' W v . Fix once for all a special point 0 ∈ A(S).
W ' T o Wx with T ⊂ Tb and Wx ' W
8.2.1.4. Parahoric and parabolic subgroups. For every α ∈ Φ, 1 6= u ∈ Uα (Qp ) there exists a unique
φα (u) ∈ R such that H(u) = α−1 (−φα (u)). Set φα (1) = +∞. This defines a map φα : Uα (Qp ) →
] − ∞, +∞] with the property that Uα,λ := φ−1
α ([λ, +∞]) ⊂ Uα (Qp ) is a subgroup for every λ ∈ R. For
every x ∈ A(S), let Px ⊂ G(Qp ) denote the subgroup generated by Z0 and the Uα,−α(x) , α ∈ Φ. When
F = C is a chamber, P0 (C) is called an Iwahori subgroup. For every facet F ⊂ A(S) define the parahoric
subgroup of type F to be the subgroup P0 (F ) ⊂ G(Qp ) generated by Z0 and the Uα,λ , α ∈ Φ, λ ∈ R such
that F ⊂ φ−1
α ([−λ, +∞]). Also let NF ⊂ N denote the pointwise stabilizer of F in N . Then the group
P (F ) := NF P0 (F ) ⊂ G(Qp ) is called the parabolic subgroup of type F .
8.2.1.5. The building I a (G, Qp ). Let I a (G, Qp ) denote the quotient of G(Qp ) × A(S) by the equivalence
relation (g, x) ∼ (g 0 , x0 ) if and only if there exists n ∈ N such that x0 = ν(n)x and g −1 g 0 n ∈ Px Nx . The
action of G(Qp ) by left multiplication on the first factor of G(Qp ) × A(S) induces an action on I a (G, Qp ).
And one shows that I a (G, Qp ) is an euclidean building - the Bruhat-Tits building of G over Qp - with
set of apartments gA(S), g ∈ G(Qp ) and set of facets gF(S), g ∈ G(Qp ). The pointwise stabilizer of a
facet gF is gP (F )g −1 , the stabilizer of gA(S) is gN g −1 and the pointwise stabilizer of gA(S) is gZ0 g −1 .
(See [T79, §3.4, 3.5]) Let F be a facet in I a (G, Qp ) and let G(Qp )F ⊂ G(Qp ) denote the stabilizer of F in
G(Qp ). Then there exists a unique smooth affine group scheme GF over Zp with generic fiber G and with
the property that GF (Ok ) = G(k)F for every finite unramified extension k/Qp (here we implicitly identify
red := G ◦
I a (G, Qp ) with its image in I a (G, k)). Write GF,F
F,Fp /Ru (GF,Fp ) for the connected reductive part
p
red is quasi-split. The link of F in
of the reduction modulo p of GF ; as G is residually quasi-split, GF,F
p
red . In particular, when F is of codimension 1, G red has semi
I a (G, Qp ) is the spherical building of GF,F
F,Fp
p
simple Fp -rank 1 and its spherical building is 0-dimensional with vertices corresponding to its minimal
parabolic subgroups. More precisely, let R denote the canonical set of generators of W (the reflexions
red /P ,
with respect to the walls of a chamber). If F is of type R \ {r}, let dr denote the dimension of GF,F
p
where P is a minimal parabolic subgroup. Then the number of vertices in the link of F is pdr + 1. The
classification shows that dr = 1, 2, 3 but we will only need the fact that dr ≥ 1. This number can also be
interpreted as the number of chambers in I a (G, Qp ) containing F that is6, degB (r) + 1.
8.2.1.6. Non type-preserving automorphisms. The action of G(Qp ) on I a (G, Qp ) is strongly transitive
(see footnote 6) but not type-preserving in general. More precisely, let G◦ ⊂ G(Qp ) denote the subgroup
acting on I a (G, Qp ) by type-preserving automorphisms; this is the subgroup generated by N◦ := ν −1 (W )
and the Uα (Qp ), α ∈ Φ. The action of G◦ on I a (G, Qp ) remains strongly transitive ( [G13, §17.7]). Set
B := P0 (C) for a chamber C in A(S); note that B ⊂ G◦ . Then (G◦ , B, N◦ ) is the Tits system induced
by the strongly transitive, type-preserving action of G◦ on I a (G, Qp ) ( [G13, §5.2]); in particular we get
the standard Bruhat-Tits decompositions [G13, §5.1-5.4]. The formalism of Tits systems (or BN -pairs)
extends (formally) to the ‘generalized’ Tits system (G(Qp ), B, N ). This is explained in [Bou68, IV, §2,
Ex. 8], [G13, §5.5, §14.7] (see also [Bo76, §3.1] and [T79, §2.5]), which we briefly summarize.
6This follows from the fact that the action of G(Q ) on I a (G, Q ) is strongly transitive (that is, transitive on the set of
p
p
embeddings of a chamber into an apartment): If C and rC are two chambers in a given apartment A with wall F and if
C 0 is another chamber in I a (G, Qp ) containing F then there exists an apartment A0 containing C, C 0 and g ∈ G(Qp ) such
that gA = A0 , gC = C. So g ∈ B. Also, the chambers C = gC and grC have the same wall gF = F in A0 , which forces
grC = C 0 . This shows that the chambers containing F in I a (G, Qp ) are C and the chambers brC, b ∈ B.
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
27
c ) Set Z◦ := Z ∩ G◦ . Since N◦ = ν −1 (W ), we have W ' N◦ /B ∩ N◦ and the following
(1) (Structure of W
commutative exact diagram
1
1
1
/ T◦ := Z◦ /B ∩ N◦
/ W = N◦ /B ∩ N◦
/ Wv
1
/ Tb = Z/B ∩ N
c = N/B ∩ N
/W
/ Wv
Ψ := Z/Z◦
'
/1
'
/1
/ Ψ := Z/Z◦
1
1
c = W o Ψ and an explicit complement of W
Furthermore the left vertical extension splits hence W
c
in W is
NC (B ∩ N )
P (C) ∩ N
'
.
Ψ=
B∩N
B∩N
Remark 8.9. The order of Ψ is bounded from above by a constant which only depends on the
Coxeter diagram ∆(W, R) of (W, R). Indeed, Ψ injects into Aut(∆(W, S)) and stabilizes the connected
components of ∆(W, R). In our case, we can also show that Ψ is Abelian. This follows for instance
from the explicit description given in [T79, §2.5]. Namely, if G0 → G denotes the simply connected
cover of G, S 0 ⊂ G0 a Qp -split maximal torus mapping into S, Z 0 := CenG0 (S 0 )(Qp ) and Z 00 :=
φ(Qp )(Z 0 ) then Ψ ' Z/Z 00 Z0 . In particular, a rough upper bound for |Ψ| is |Ψ| ≤ ι(|µG |).
(2) (Double cosets) The map ŵ → B ŵB induces a bijection
c →B
W
˜ \ G(Qp )/B.
(3) (Subgroups of G(Qp ) containing B) Recall that R denotes the canonical set of generators of W and
let R denote the set of pairs (Ψ0 , R0 ) with Ψ0 ⊂ Ψ a subgroup and R0 ⊂ R a subset normalized by
Ψ0 . For a subset R0 ⊂ R, let WR0 ⊂ W denote the subgroup generated by R0 ; if R0 ( R the group
WR0 is finite. Then the map
G
(Ψ0 , R0 ) → P(Ψ0 ,R0 ) := BWR0 Ψ0 B :=
B ŵB
ŵ∈WR0 Ψ0
induces a bijection from R to the set of subgroups of G(Qp ) containing B.
Furthermore a subgroup P(Ψ0 ,R0 ) ( G(Qp ) is a maximal compact subgroup if and only if R0 ( R,
Ψ0 = NorΨ (R0 ) and Ψ0 acts transitively on R \ R0 .
8.2.2. Proof of Lemma 8.6. Let ϕ : G0 → G denote the simply connected cover of G. The exact sequence
(8.3)
1 → µG (Qp ) → G0 (Qp ) → G(Qp ) → H1 (Qp , µG (Qp ))
shows that ϕ : G0 (Qp ) → G(Qp ) has finite kernel and finite cokernel. So, from Lemma 8.4, we may and
will assume that G is simply connected and that K = B ⊂ G(Qp ) is an Iwahori subgroup. Then from
Subsection 8.2.1.6 (3) we have the Bruhat-Tits decomposition
G
G(Qp ) =
BwB
w∈W
As (W, R) is a Coxeter system, every element w ∈ W can be written as
w = r1 · · · r`(w)
with r1 , . . . , r`(w) ∈ R and `(w) ∈ Z≥0 minimal (called the length of w) (the elements r1 , . . . , r` (w) are
then unique up to permutation). Furthermore, we have
degB (w) = degB (r1 ) · · · degB (r`(w) )
28
ANNA CADORET AND ARNO KRET
with degB (r) = pdr and dr = 1, 2, 3 for every r ∈ R (See Paragrah 8.2.1.5). In particular, degB (w) ≥ p`(w) .
The conclusion then follows from the fact that there are only finitely many elements of bounded length
in a Coxeter group.
8.2.3. Proof of Lemma 8.8. From Subsection 8.2.1.6 (3), K ⊂ G(Qp ) is of the form
G
K=
B ŵB
ŵ∈WR0 Ψ0
0
0
0
0
0
for a subset R0 ( R and a subgroup
GΨ ⊂ Ψ such that Ψ = NorΨ (R ) and Ψ acts transitively on R \ R .
cK := WR0 Ψ0 and K◦ :=
Write W
BwB (so that K = K◦ Ψ0 = Ψ0 K◦ ). Then,
w∈WR0
Lemma 8.10. We have
G
G(Qp ) =
K wK.
b
cK \W
c /W
cK
ŵ∈W
Proof. Indeed, again from Subsection 8.2.1.6 (3) we have the Bruhat-Tits decomposition
G
G
G
cK ŵW
cK B =
G(Qp ) =
B ŵB =
BW
Ψ0 BWR0 ŵWR0 BΨ0 .
cK \W
c /W
cK
ŵ∈W
c
ŵ∈W
cK \W
c /W
cK
ŵ∈W
Also, observe that as Ψ0 stabilizes R0 , R, it acts by conjugation on WR0 , W and the canonical surjection
cK \ W
c /W
cK
WR0 \ W/WR0 W
induces a bijection
cK \ W
c /W
cK .
(WR0 \ W/WR0 )/Ψ0 →
˜W
As a result, we have
G(Qp ) =
G
G
Ψ0 BWR0 wWR0 BΨ0 =
Ψ0 K◦ wK◦ Ψ0 .
w∈(WR0 \W/WR0 )/Ψ0
w∈(WR0 \W/WR0 )/Ψ0
Here, the equality
K◦ wK◦ = BWR0 wWR0 B
is a formal consequence of the Bruhat-Tits decomposition for the Tits system (G◦ , B, N◦ ) (See [G13, §5.4]).
As G(Qp ) =
G
c we have
B ŵB, Lemma 8.10 also shows that for every ŵ ∈ W
c
ŵ∈W
G
cK ŵW
cK B =
K ŵK = B W
BλB.
cK ŵW
cK
λ∈W
As a result,
X
|B \ K ŵK| =
degB (λ).
cK ŵW
cK
λ∈W
But, also,
|B \ K ŵK| = degK (ŵ)[K : B] = degK (ŵ)
X
degB (µ).
cK
µ∈W
From this, we get
P
degK (ŵ) =
cK ŵW
cK degB (λ)
λ∈W
P
.
c degB (µ)
µ∈W
K
Now, we can compute explicitly
X
X X
degB (µ) =
degB (w) = |Ψ|(1 +
cK
µ∈W
ψ∈Ψ w∈WR0
X
16=w∈WR0
Write
cK \ W
cK ŵW
cK = S1 t S2 ,
W
degB (w)) ≡ |Ψ| [p].
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
29
cK λ such that W
cK λ ∩ Ψ 6= ∅. Then, for C ∈ S1 we have
where S1 denote the set of left cosets W
X
X
degB (λ) =
degB (λ)
λ∈C
cK
λ∈W
hence
P
degK (ŵ) = |S1 | +
C∈C2
P
degB (λ)
.
degB (µ)
λ∈C
P
cK
µ∈W
But for C ∈ S2 and λ ∈ C we have λ = wψ withP1 6= w ∈ W and ψ ∈ Ψ. In particular, degB (λ) = degB (w)
is a power of p. As degK (ŵ) is an integer and µ∈W
cK degB (µ) ≡ |Ψ| [p] is non-zero (this is where we use
cK then S2 6= ∅. This follows
the assumption p > ι(|µG |)(≥ |Ψ|)), it is thus enough to prove that if ŵ ∈
/W
from the maximality of K. Indeed, otherwise, we may assume that ŵ = ψ ∈ Ψ \ Ψ0 . Then the following
holds
cK there exists ψ(w, ψ) ∈ Ψ \ Ψ0 , w(w, ψ) ∈ W
cK such that ψw = w(w, ψ)ψ(w, ψ).
(8.4) for every w ∈ W
But, then,
cK ψ(w, ψ)W
cK = W
cK ψ W
cK
W
cK and ψ in W
c is of the
thus ψ(w, ψ) satisfies again (8.4). This shows that the subgroup generated by W
00
0
00
c
form WK o Ψ for some subgroup Ψ ( Ψ ⊂ Ψ, which contradicts the maximality of K. 8.3. Proof of Theorem 7.3. Recall that we may assume that Γ = G(Q) ∩ K for some compact open
subgroup K as in the proof of Proposition 8.5 and that Γ− ⊂ G(Af ) denote the closure of Γ for the adelic
topology.
Lemma 8.11. For every a ∈ G(Q), the canonical map ϕa : Γ\ΓaΓ → Γ− \Γ− aΓ− is bijective. In particular degΓ (a) = degΓ− (a).
Proof. We first prove that ϕa is injective. Let Γaγ, Γaγ 0 ∈ Γ\ΓaΓ such that Γ− aγ = Γ− aγ 0 that is there
0
exists γ − ∈ Γ− such that aγ = γ − aγ 0 . But, then γ − = aγγ −1 a−1 ∈ Γ− ∩ G(Q) = Γ. Thus Γaγ = Γaγ 0 .
We now show that ϕa is surjective. As ϕa : Γ \ ΓaΓ ,→ Γ− \ Γ− aΓ− is injective and both sets are finite,
it is enough to prove that degΓ− (a) ≤ degΓ (a). For this, just write
G
ΓaΓ =
Γai
1≤i≤degΓ (a)
and observe that
G
Γ− aΓ− = (ΓaΓ)− = (
Γai )− =
1≤i≤degΓ (a)
[
(Γai )− =
1≤i≤degΓ (a)
[
Γ− ai .
1≤i≤degΓ (a)
Here, the first equality follows from the fact that Γ− is compact (to prove that (ΓaΓ)− ⊂ Γ− aΓ− ).
Lemma 8.12. The canonical map ϕ : Γ\G(Q)/Γ → K\G(Af )/K has finite fibers.
Proof. Let a ∈ G(Q) and let R ⊂ G(Q) be a set of representatives for ϕ−1 (KaK). Since Γ = G(Q) ∩ K,
the map Γ\G(Q) → K\G(Af ) is injective hence restricts to an injective map
G
G
Γ\
ΓbΓ =
Γ\ΓbΓ −→ K\KaK.
b∈R
b∈R
Because the union is disjoint, we get
G
|R| ≤ Γ\ΓbΓ ≤ |K\KaK| = degK (a).
b∈R
30
ANNA CADORET AND ARNO KRET
For simply connected groups G of non-compact type, the proof is now complete. Indeed, in that case,
by strong approximation, Γ− = K. So the conclusion follows from the combination of Lemma 8.11,
Proposition 8.5 and Lemma 8.12. In the non-simply connected case, we can no longer apply Lemma 8.11
directly.
Q
Lemma 8.13. There exists an integer r ≥ 1 and a compact normal subgroup H = p Hp ⊂ K, with
Hp ⊂ G(Qp ) compact open, [Kp : Hp ] ≤ r and such that Γ− , H ⊂ G(Af ) are commensurable.
Proof. Let ϕ : G0 → G denote the simply connected cover of G and let µG denote its kernel, which is a finite
group of multiplicative type. Let P denote a finite set of primes
Qcontaining the primes where G ramifies
and the primes < |µG |. Fix a compact open subgroup K 0 = p Kp0 ⊂ G0 (Af ), with Kp0 ⊂ ϕ−1 (Kp ) ⊂
G0 (Qp ) a compact open subgroup and Kp0 ⊂ Q
G(Qp ) hyperspecial for p ∈
/ P. Set Γ0 = K 0 ∩ G0 (Q). We take
0
0
H := ϕ(K ). Then, by construction, H = p Hp , with Hp = ϕ(Kp ) ⊂ Kp ⊂ G(Qp ) open. For p ∈ P,
replace Hp with CorKp (Hp ), which is a normal compact open subgroup of Kp . Up to increasing P, we
may assume that for p ∈
/ P, the short exact sequence
ϕ
1 → µG → G0 → G → 1
extends to a short exact sequence of smooth group schemes over Zp with Kp0 = G0 (Zp ), Kp = G(Zp ).
Taking the reduction modulo p and Fp -points, we obtain a short exact sequence of finite groups
ϕ
1 → µG (Fp ) → G0 (Fp ) → G(Fp ) → 1.
ur
Since Zur
p (the integral closure of Zp in the maximal unramified extension Qp of Qp ) is Henselian, by
smoothness (see [PlR94, Lemma 6.5, p. 295] for details) we get a short exact sequence
ϕ
0
ur
ur
1 → µG (Zur
p ) → G (Zp ) → G(Zp ) → 1.
Taking the Gal(Qur
p /Qp )-invariants, we obtain
ur
Hp = ker(Kp → H1 (Gal(Qur
p /Qp ), µ(Zp ))).
In particular, Hp is normal in Kp (thus H is normal in K as required) and
ur
[Kp : Hp ] ≤ |H1 (Gal(Qur
p /Qp ), µG (Zp ))| ≤ |µG |.
(8.5)
The last inequality comes from the fact that Gal(Qur
p /Qp ) ' Ẑ is procyclic hence ( [S68, Chap. XIII, §1,
Prop. 1])
ur
ur
H1 (Gal(Qur
p /Qp ), µG (Zp )) = µG (Zp )Gal(Qur
p /Qp )
ur
(the maximal trivial Gal(Qur
p /Qp )-quotient of µG (Zp )). Thus we can take
r := max{[Kp : Hp ] | p ∈ P} ∪ {ι(|µG |)}.
Γ− , H
It remains to show that
⊂ G(Af ) are commensurable. As Γ0 ⊂ G0 (Q) is arithmetic and ϕ : G0 → G
is surjective, the group ϕ(Γ0 ) ⊂ G(Q) is again arithmetic ( [PlR94, Thm. 4.1]) hence [Γ : ϕ(Γ0 )] is finite.
0
0
Thus [Γ− : ϕ(Γ ,− )](≤ [Γ : Γ0 ]) is finite as well. But, by strong approximation, Γ − = K 0 and by the
0
continuity of ϕ and the compacity of Γ − ,
0
ϕ(Γ0 )− = ϕ(Γ − ) = ϕ(K 0 ) = H.
Lemma 8.14. For every integer d ≥ 1 there are only finitely double-classes KaK ∈ K\G(Q)/K with
degΓ− (a) ≤ d.
Proof.
Let a ∈ G(Q) with degΓ− (a) ≤ d and let H be as in Lemma 8.13. Then dCH,Γ− ≥ degH (a) =
Q
deg
/ P, p > ι(|µG |)rdCH,Γ− , we
Hp (a) and degHp (a) ≥ degKp (a)/r (Lemma 8.2). In particular, for p ∈
p
have a ∈ Kp (Lemma 8.8). Set
N = N (G, Γ− , H, d) := |P| + |{p ∈
/ P, p ≤ drCH,Γ− ι(|µG |)}|
Then (Lemma 8.2),
degK (a) =
Y
p∈P
degKp (a)
Y
p∈P,
/ p≤drCH,Γ− ι(|µG |)
degKp (a) ≤ rN degH (a) ≤ rN dCH,Γ− .
GALOIS-GENERIC POINTS ON CONNECTED SHIMURA VARIETIES
The conclusion thus follows from Proposition 8.5.
31
End of the proof of Proposition 7.3: Let a ∈ G(Q) such that degΓ (a) ≤ d. Then degΓ− (a) =
degΓ (a) ≤ d (Lemma 8.11). So there are only finitely many possibilities for the set of double-classes
KaK ∈ K\G(Q)/K (Lemma 8.14) hence only finitely many possibilities for the set of double-classes
ΓaΓ ∈ Γ\G(Q)/Γ (Lemma 8.12).
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Centre de Mathématiques Laurent Schwartz - Ecole Polytechnique,
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